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Properties

Label 2-15e2-225.77-c1-0-12
Degree $2$
Conductor $225$
Sign $0.384 - 0.923i$
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  + (1.03 − 0.397i)2-s + (0.642 + 1.60i)3-s + (−0.572 + 0.515i)4-s + (−0.337 + 2.21i)5-s + (1.30 + 1.40i)6-s + (0.651 − 2.43i)7-s + (−1.39 + 2.73i)8-s + (−2.17 + 2.06i)9-s + (0.529 + 2.42i)10-s + (0.992 − 2.23i)11-s + (−1.19 − 0.589i)12-s + (−0.0280 + 0.0731i)13-s + (−0.291 − 2.77i)14-s + (−3.77 + 0.878i)15-s + (−0.195 + 1.85i)16-s + (5.10 + 2.60i)17-s + ⋯
L(s)  = 1  + (0.732 − 0.281i)2-s + (0.371 + 0.928i)3-s + (−0.286 + 0.257i)4-s + (−0.150 + 0.988i)5-s + (0.532 + 0.575i)6-s + (0.246 − 0.918i)7-s + (−0.493 + 0.967i)8-s + (−0.724 + 0.689i)9-s + (0.167 + 0.766i)10-s + (0.299 − 0.672i)11-s + (−0.345 − 0.170i)12-s + (−0.00778 + 0.0202i)13-s + (−0.0779 − 0.741i)14-s + (−0.973 + 0.226i)15-s + (−0.0487 + 0.464i)16-s + (1.23 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42396 + 0.949873i\)
\(L(\frac12)\) \(\approx\) \(1.42396 + 0.949873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.642 - 1.60i)T \)
5 \( 1 + (0.337 - 2.21i)T \)
good2 \( 1 + (-1.03 + 0.397i)T + (1.48 - 1.33i)T^{2} \)
7 \( 1 + (-0.651 + 2.43i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.992 + 2.23i)T + (-7.36 - 8.17i)T^{2} \)
13 \( 1 + (0.0280 - 0.0731i)T + (-9.66 - 8.69i)T^{2} \)
17 \( 1 + (-5.10 - 2.60i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.99 + 0.648i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.94 + 4.86i)T + (-4.78 - 22.4i)T^{2} \)
29 \( 1 + (-5.33 + 1.13i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (2.55 + 0.542i)T + (28.3 + 12.6i)T^{2} \)
37 \( 1 + (1.17 + 7.41i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.972 - 2.18i)T + (-27.4 + 30.4i)T^{2} \)
43 \( 1 + (2.31 + 0.621i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.09 - 7.83i)T + (-19.1 - 42.9i)T^{2} \)
53 \( 1 + (11.3 - 5.76i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-8.91 + 3.97i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (9.96 + 4.43i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (2.63 + 4.06i)T + (-27.2 + 61.2i)T^{2} \)
71 \( 1 + (-5.52 - 1.79i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.05 - 12.9i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (1.55 + 7.30i)T + (-72.1 + 32.1i)T^{2} \)
83 \( 1 + (1.34 - 0.0702i)T + (82.5 - 8.67i)T^{2} \)
89 \( 1 + (-6.60 + 4.80i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (16.1 + 10.5i)T + (39.4 + 88.6i)T^{2} \)
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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

−12.45582867836846578757388083318, −11.24204324991696253816596671017, −10.74742791200155879531414523992, −9.695330193480095722015320796442, −8.463887700129384954007514630635, −7.57758005010277320616534135878, −6.00710921393648012578447382171, −4.68997628581308697101834032882, −3.69135089566902957004148114685, −2.95683088419615055178849191299, 1.35416520778900689034672611571, 3.32181232492011291253001077912, 4.96140471090836610817001922479, 5.61364172367931342015438149472, 6.90516197163401326230314907885, 8.064210066435615099040626328517, 9.064262799769130097403861651733, 9.741786450525176131079525351792, 11.87545096652118668819474727420, 12.13754039847261576250497601341

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