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Properties

Label 2-63-7.4-c5-0-4
Degree $2$
Conductor $63$
Sign $-0.984 + 0.174i$
Analytic cond. $10.1041$
Root an. cond. $3.17870$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.11 + 8.85i)2-s + (−36.2 + 62.8i)4-s + (11.8 + 20.5i)5-s + (30.6 + 125. i)7-s − 414.·8-s + (−121. + 210. i)10-s + (232. − 403. i)11-s − 1.01e3·13-s + (−958. + 915. i)14-s + (−957. − 1.65e3i)16-s + (280. − 486. i)17-s + (693. + 1.20e3i)19-s − 1.72e3·20-s + 4.75e3·22-s + (2.05e3 + 3.56e3i)23-s + ⋯
L(s)  = 1  + (0.903 + 1.56i)2-s + (−1.13 + 1.96i)4-s + (0.212 + 0.367i)5-s + (0.236 + 0.971i)7-s − 2.28·8-s + (−0.383 + 0.665i)10-s + (0.580 − 1.00i)11-s − 1.67·13-s + (−1.30 + 1.24i)14-s + (−0.935 − 1.61i)16-s + (0.235 − 0.408i)17-s + (0.440 + 0.763i)19-s − 0.963·20-s + 2.09·22-s + (0.810 + 1.40i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.984 + 0.174i$
Analytic conductor: \(10.1041\)
Root analytic conductor: \(3.17870\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :5/2),\ -0.984 + 0.174i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.212439 - 2.41369i\)
\(L(\frac12)\) \(\approx\) \(0.212439 - 2.41369i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-30.6 - 125. i)T \)
good2 \( 1 + (-5.11 - 8.85i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-11.8 - 20.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-232. + 403. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.01e3T + 3.71e5T^{2} \)
17 \( 1 + (-280. + 486. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-693. - 1.20e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.05e3 - 3.56e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.38e3T + 2.05e7T^{2} \)
31 \( 1 + (1.47e3 - 2.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.95e3 - 8.58e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 4.47e3T + 1.15e8T^{2} \)
43 \( 1 - 5.18e3T + 1.47e8T^{2} \)
47 \( 1 + (1.56e3 + 2.70e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-570. + 988. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.37e4 + 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.05e4 - 1.82e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.77e4 + 4.81e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.07e3T + 1.80e9T^{2} \)
73 \( 1 + (-8.38e3 + 1.45e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.42e3 - 4.19e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.01e4T + 3.93e9T^{2} \)
89 \( 1 + (3.12e4 + 5.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 6.36e4T + 8.58e9T^{2} \)
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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

−14.52998048216707017132261711824, −13.98362986695719015257406247753, −12.60242287644634859809301097199, −11.66297364727200424223477070071, −9.529528061078928753942223640790, −8.273118453698472296274698501049, −7.10360531170098358735588529716, −5.89551399692766232195554371632, −4.94743286478410068285764423294, −3.09802947825728272826189930714, 0.900750391196867634479850628379, 2.44655853350267015015861280444, 4.22625444851309139243255449987, 5.04437905922265098265754137820, 7.14391194419255722606568844502, 9.338492493681443476343619835912, 10.18020752216039981552873494098, 11.24976790682755540012549660378, 12.42402010727372423110855778097, 12.99533326876130253629722119211

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