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Properties

Label 2-15e2-225.106-c1-0-24
Degree 22
Conductor 225225
Sign 0.999+0.0258i0.999 + 0.0258i
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  + (2.43 + 0.516i)2-s + (0.810 − 1.53i)3-s + (3.81 + 1.69i)4-s + (−2.22 + 0.247i)5-s + (2.76 − 3.30i)6-s + (−0.644 + 1.11i)7-s + (4.36 + 3.17i)8-s + (−1.68 − 2.48i)9-s + (−5.52 − 0.546i)10-s + (0.502 + 0.106i)11-s + (5.68 − 4.45i)12-s + (−4.87 + 1.03i)13-s + (−2.14 + 2.38i)14-s + (−1.42 + 3.60i)15-s + (3.39 + 3.76i)16-s + (5.60 + 4.07i)17-s + ⋯
L(s)  = 1  + (1.71 + 0.365i)2-s + (0.467 − 0.883i)3-s + (1.90 + 0.848i)4-s + (−0.993 + 0.110i)5-s + (1.12 − 1.34i)6-s + (−0.243 + 0.421i)7-s + (1.54 + 1.12i)8-s + (−0.562 − 0.827i)9-s + (−1.74 − 0.172i)10-s + (0.151 + 0.0321i)11-s + (1.64 − 1.28i)12-s + (−1.35 + 0.287i)13-s + (−0.572 + 0.636i)14-s + (−0.367 + 0.930i)15-s + (0.847 + 0.941i)16-s + (1.35 + 0.987i)17-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((0.999+0.0258i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0258i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((0.999+0.0258i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 0.999+0.0258i0.999 + 0.0258i
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), 0.999+0.0258i)(2,\ 225,\ (\ :1/2),\ 0.999 + 0.0258i)

Particular Values

L(1)L(1) ≈\approx 2.84799−0.0368520i2.84799 - 0.0368520i
L(12)L(\frac12) ≈\approx 2.84799−0.0368520i2.84799 - 0.0368520i
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+(−0.810+1.53i)T 1 + (-0.810 + 1.53i)T
5 1+(2.22−0.247i)T 1 + (2.22 - 0.247i)T
good2 1+(−2.43−0.516i)T+(1.82+0.813i)T2 1 + (-2.43 - 0.516i)T + (1.82 + 0.813i)T^{2}
7 1+(0.644−1.11i)T+(−3.5−6.06i)T2 1 + (0.644 - 1.11i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.502−0.106i)T+(10.0+4.47i)T2 1 + (-0.502 - 0.106i)T + (10.0 + 4.47i)T^{2}
13 1+(4.87−1.03i)T+(11.8−5.28i)T2 1 + (4.87 - 1.03i)T + (11.8 - 5.28i)T^{2}
17 1+(−5.60−4.07i)T+(5.25+16.1i)T2 1 + (-5.60 - 4.07i)T + (5.25 + 16.1i)T^{2}
19 1+(4.21+3.06i)T+(5.87+18.0i)T2 1 + (4.21 + 3.06i)T + (5.87 + 18.0i)T^{2}
23 1+(−2.78+3.09i)T+(−2.40−22.8i)T2 1 + (-2.78 + 3.09i)T + (-2.40 - 22.8i)T^{2}
29 1+(−0.724−6.89i)T+(−28.3+6.02i)T2 1 + (-0.724 - 6.89i)T + (-28.3 + 6.02i)T^{2}
31 1+(−0.160+1.52i)T+(−30.3−6.44i)T2 1 + (-0.160 + 1.52i)T + (-30.3 - 6.44i)T^{2}
37 1+(−2.27+6.99i)T+(−29.9−21.7i)T2 1 + (-2.27 + 6.99i)T + (-29.9 - 21.7i)T^{2}
41 1+(3.98−0.847i)T+(37.4−16.6i)T2 1 + (3.98 - 0.847i)T + (37.4 - 16.6i)T^{2}
43 1+(−2.08+3.60i)T+(−21.5−37.2i)T2 1 + (-2.08 + 3.60i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.841−8.00i)T+(−45.9+9.77i)T2 1 + (-0.841 - 8.00i)T + (-45.9 + 9.77i)T^{2}
53 1+(−2.21+1.60i)T+(16.3−50.4i)T2 1 + (-2.21 + 1.60i)T + (16.3 - 50.4i)T^{2}
59 1+(−2.99+0.636i)T+(53.8−23.9i)T2 1 + (-2.99 + 0.636i)T + (53.8 - 23.9i)T^{2}
61 1+(7.17+1.52i)T+(55.7+24.8i)T2 1 + (7.17 + 1.52i)T + (55.7 + 24.8i)T^{2}
67 1+(−0.0475+0.452i)T+(−65.5−13.9i)T2 1 + (-0.0475 + 0.452i)T + (-65.5 - 13.9i)T^{2}
71 1+(−1.33+0.971i)T+(21.9−67.5i)T2 1 + (-1.33 + 0.971i)T + (21.9 - 67.5i)T^{2}
73 1+(2.72+8.38i)T+(−59.0+42.9i)T2 1 + (2.72 + 8.38i)T + (-59.0 + 42.9i)T^{2}
79 1+(−0.359−3.42i)T+(−77.2+16.4i)T2 1 + (-0.359 - 3.42i)T + (-77.2 + 16.4i)T^{2}
83 1+(−13.4+5.97i)T+(55.5−61.6i)T2 1 + (-13.4 + 5.97i)T + (55.5 - 61.6i)T^{2}
89 1+(2.98+9.17i)T+(−72.0+52.3i)T2 1 + (2.98 + 9.17i)T + (-72.0 + 52.3i)T^{2}
97 1+(0.842+8.01i)T+(−94.8+20.1i)T2 1 + (0.842 + 8.01i)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

−12.43003902304948699130454572461, −12.01939567263003417093220662269, −10.82726883567739543873980652179, −8.973106206397156242413778582474, −7.73694067930769154802988579671, −7.04650616225401601872798076732, −6.10276857816297798738335548767, −4.78238138020432126889965660853, −3.55852617082274613457225796128, −2.52413245299591692806872988974, 2.77967266079489783154697302563, 3.69207246515493831255320803557, 4.59531327440372357915460960210, 5.45121230527446554362001574851, 7.06752180575170395647437507830, 8.082058167993191850267601669546, 9.726255269727761613207964181992, 10.53532582866018368403945086351, 11.67372760538541490156310352368, 12.16469950374818930746153166182

Graph of the ZZ-function along the critical line