Properties

Degree $2$
Conductor $63$
Sign $0.549 + 0.835i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 − 1.47i)2-s + (0.349 + 1.69i)3-s + (−0.444 + 0.769i)4-s + (1.79 − 3.10i)5-s + (2.19 − 1.95i)6-s + (0.5 + 0.866i)7-s − 1.88·8-s + (−2.75 + 1.18i)9-s − 6.09·10-s + (1.40 + 2.43i)11-s + (−1.46 − 0.484i)12-s + (−0.5 + 0.866i)13-s + (0.849 − 1.47i)14-s + (5.89 + 1.95i)15-s + (2.49 + 4.31i)16-s − 4.11·17-s + ⋯
L(s)  = 1  + (−0.600 − 1.04i)2-s + (0.201 + 0.979i)3-s + (−0.222 + 0.384i)4-s + (0.802 − 1.38i)5-s + (0.897 − 0.798i)6-s + (0.188 + 0.327i)7-s − 0.667·8-s + (−0.918 + 0.395i)9-s − 1.92·10-s + (0.423 + 0.733i)11-s + (−0.421 − 0.139i)12-s + (−0.138 + 0.240i)13-s + (0.227 − 0.393i)14-s + (1.52 + 0.505i)15-s + (0.623 + 1.07i)16-s − 0.997·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.549 + 0.835i$
Motivic weight: \(1\)
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :1/2),\ 0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678233 - 0.365925i\)
\(L(\frac12)\) \(\approx\) \(0.678233 - 0.365925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.349 - 1.69i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.849 + 1.47i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.11T + 17T^{2} \)
19 \( 1 + 0.888T + 19T^{2} \)
23 \( 1 + (2.93 - 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.849 - 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.49 + 6.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.76T + 37T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.60 + 4.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 - 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.123T + 53T^{2} \)
59 \( 1 + (-4.43 + 7.68i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.15 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.87T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (-3.54 - 6.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.05 - 3.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.60T + 89T^{2} \)
97 \( 1 + (3.66 + 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.92863901539881217496686136701, −13.52002562117593653564472894820, −12.28737090580311813368314273210, −11.30450476122182957355114125737, −9.920919263705025788097436128276, −9.348070393168372288304057211402, −8.533101499363056205200552398851, −5.76824547598489157083295571479, −4.37226227371296963193459256221, −2.07543646350802292275723653096, 2.77665001478225031152817436608, 6.18579846603422312459981074019, 6.63714603147415688667459483870, 7.81739185740669098098179593972, 8.959962920203876046251110770482, 10.48996833534964353614124282230, 11.75495578923460170695847917669, 13.34530560228949481007928925304, 14.30471593546078232965576288278, 14.92954095398782106795011851163

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