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Properties

Label 2-15e2-225.106-c1-0-7
Degree 22
Conductor 225225
Sign −0.190−0.981i-0.190 - 0.981i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 + 0.380i)2-s + (−1.49 + 0.871i)3-s + (1.23 + 0.551i)4-s + (−0.0204 + 2.23i)5-s + (−3.01 + 0.991i)6-s + (−2.15 + 3.73i)7-s + (−0.953 − 0.692i)8-s + (1.48 − 2.60i)9-s + (−0.888 + 3.99i)10-s + (0.885 + 0.188i)11-s + (−2.33 + 0.253i)12-s + (6.04 − 1.28i)13-s + (−5.29 + 5.87i)14-s + (−1.91 − 3.36i)15-s + (−3.26 − 3.62i)16-s + (4.65 + 3.38i)17-s + ⋯
L(s)  = 1  + (1.26 + 0.269i)2-s + (−0.864 + 0.503i)3-s + (0.619 + 0.275i)4-s + (−0.00914 + 0.999i)5-s + (−1.23 + 0.404i)6-s + (−0.816 + 1.41i)7-s + (−0.337 − 0.244i)8-s + (0.493 − 0.869i)9-s + (−0.280 + 1.26i)10-s + (0.266 + 0.0567i)11-s + (−0.674 + 0.0733i)12-s + (1.67 − 0.356i)13-s + (−1.41 + 1.57i)14-s + (−0.495 − 0.868i)15-s + (−0.815 − 0.905i)16-s + (1.12 + 0.819i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.190−0.981i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.190−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.190−0.981i-0.190 - 0.981i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(106,⋅)\chi_{225} (106, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), −0.190−0.981i)(2,\ 225,\ (\ :1/2),\ -0.190 - 0.981i)

Particular Values

L(1)L(1) ≈\approx 1.01017+1.22454i1.01017 + 1.22454i
L(12)L(\frac12) ≈\approx 1.01017+1.22454i1.01017 + 1.22454i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(1.49−0.871i)T 1 + (1.49 - 0.871i)T
5 1+(0.0204−2.23i)T 1 + (0.0204 - 2.23i)T
good2 1+(−1.79−0.380i)T+(1.82+0.813i)T2 1 + (-1.79 - 0.380i)T + (1.82 + 0.813i)T^{2}
7 1+(2.15−3.73i)T+(−3.5−6.06i)T2 1 + (2.15 - 3.73i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.885−0.188i)T+(10.0+4.47i)T2 1 + (-0.885 - 0.188i)T + (10.0 + 4.47i)T^{2}
13 1+(−6.04+1.28i)T+(11.8−5.28i)T2 1 + (-6.04 + 1.28i)T + (11.8 - 5.28i)T^{2}
17 1+(−4.65−3.38i)T+(5.25+16.1i)T2 1 + (-4.65 - 3.38i)T + (5.25 + 16.1i)T^{2}
19 1+(2.14+1.56i)T+(5.87+18.0i)T2 1 + (2.14 + 1.56i)T + (5.87 + 18.0i)T^{2}
23 1+(−2.35+2.61i)T+(−2.40−22.8i)T2 1 + (-2.35 + 2.61i)T + (-2.40 - 22.8i)T^{2}
29 1+(−0.238−2.26i)T+(−28.3+6.02i)T2 1 + (-0.238 - 2.26i)T + (-28.3 + 6.02i)T^{2}
31 1+(−0.244+2.32i)T+(−30.3−6.44i)T2 1 + (-0.244 + 2.32i)T + (-30.3 - 6.44i)T^{2}
37 1+(0.704−2.16i)T+(−29.9−21.7i)T2 1 + (0.704 - 2.16i)T + (-29.9 - 21.7i)T^{2}
41 1+(−5.91+1.25i)T+(37.4−16.6i)T2 1 + (-5.91 + 1.25i)T + (37.4 - 16.6i)T^{2}
43 1+(1.16−2.00i)T+(−21.5−37.2i)T2 1 + (1.16 - 2.00i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.583+5.54i)T+(−45.9+9.77i)T2 1 + (0.583 + 5.54i)T + (-45.9 + 9.77i)T^{2}
53 1+(−3.50+2.54i)T+(16.3−50.4i)T2 1 + (-3.50 + 2.54i)T + (16.3 - 50.4i)T^{2}
59 1+(−12.8+2.73i)T+(53.8−23.9i)T2 1 + (-12.8 + 2.73i)T + (53.8 - 23.9i)T^{2}
61 1+(13.8+2.93i)T+(55.7+24.8i)T2 1 + (13.8 + 2.93i)T + (55.7 + 24.8i)T^{2}
67 1+(0.796−7.57i)T+(−65.5−13.9i)T2 1 + (0.796 - 7.57i)T + (-65.5 - 13.9i)T^{2}
71 1+(4.21−3.05i)T+(21.9−67.5i)T2 1 + (4.21 - 3.05i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.0260−0.0802i)T+(−59.0+42.9i)T2 1 + (-0.0260 - 0.0802i)T + (-59.0 + 42.9i)T^{2}
79 1+(−0.0932−0.887i)T+(−77.2+16.4i)T2 1 + (-0.0932 - 0.887i)T + (-77.2 + 16.4i)T^{2}
83 1+(−0.951+0.423i)T+(55.5−61.6i)T2 1 + (-0.951 + 0.423i)T + (55.5 - 61.6i)T^{2}
89 1+(−1.84−5.68i)T+(−72.0+52.3i)T2 1 + (-1.84 - 5.68i)T + (-72.0 + 52.3i)T^{2}
97 1+(−0.171−1.63i)T+(−94.8+20.1i)T2 1 + (-0.171 - 1.63i)T + (-94.8 + 20.1i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.56127473337965738570006098250, −11.76402643031997489085433231596, −10.84279480587660291069927958893, −9.807338170133371482101119654829, −8.700493213083455954035949567874, −6.71732257000589009935331769672, −6.07230780817438044834737312909, −5.53047307258736120499784711964, −3.92131867747127938409265032925, −3.07028475699534538771219895340, 1.11762946934577242663088876464, 3.60183300742592881802237602923, 4.44980498860464769320313272925, 5.65119696274948947437064096873, 6.45052000017065001860880743569, 7.69153775852712221401287296762, 9.109530667796306278230120802393, 10.43204185264518508502072828607, 11.38975138469424284724852361337, 12.19715894476490041318328562373

Graph of the ZZ-function along the critical line