Properties

Label 2-15e2-225.121-c1-0-27
Degree $2$
Conductor $225$
Sign $-0.190 + 0.981i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01017 - 1.22454i\)
\(L(\frac12)\) \(\approx\) \(1.01017 - 1.22454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line