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Properties

Label 2-63-9.4-c3-0-16
Degree $2$
Conductor $63$
Sign $-0.998 + 0.0460i$
Analytic cond. $3.71712$
Root an. cond. $1.92798$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 − 4.02i)2-s + (3.90 − 3.43i)3-s + (−6.78 + 11.7i)4-s + (8.24 − 14.2i)5-s + (−22.8 − 7.73i)6-s + (−3.5 − 6.06i)7-s + 25.9·8-s + (3.45 − 26.7i)9-s − 76.5·10-s + (23.3 + 40.5i)11-s + (13.8 + 69.1i)12-s + (−20.8 + 36.0i)13-s + (−16.2 + 28.1i)14-s + (−16.8 − 83.9i)15-s + (−5.85 − 10.1i)16-s − 49.9·17-s + ⋯
L(s)  = 1  + (−0.821 − 1.42i)2-s + (0.751 − 0.660i)3-s + (−0.848 + 1.46i)4-s + (0.737 − 1.27i)5-s + (−1.55 − 0.525i)6-s + (−0.188 − 0.327i)7-s + 1.14·8-s + (0.128 − 0.991i)9-s − 2.42·10-s + (0.641 + 1.11i)11-s + (0.333 + 1.66i)12-s + (−0.443 + 0.768i)13-s + (−0.310 + 0.537i)14-s + (−0.289 − 1.44i)15-s + (−0.0915 − 0.158i)16-s − 0.712·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0460i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.998 + 0.0460i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ -0.998 + 0.0460i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0279367 - 1.21172i\)
\(L(\frac12)\) \(\approx\) \(0.0279367 - 1.21172i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.90 + 3.43i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good2 \( 1 + (2.32 + 4.02i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-8.24 + 14.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-23.3 - 40.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.8 - 36.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 49.9T + 4.91e3T^{2} \)
19 \( 1 - 27.2T + 6.85e3T^{2} \)
23 \( 1 + (-93.7 + 162. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-95.7 - 165. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-30.1 + 52.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 82.4T + 5.06e4T^{2} \)
41 \( 1 + (-114. + 197. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-163. - 282. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (268. + 465. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 554.T + 1.48e5T^{2} \)
59 \( 1 + (241. - 419. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-365. - 632. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (171. - 297. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 231.T + 3.57e5T^{2} \)
73 \( 1 - 420.T + 3.89e5T^{2} \)
79 \( 1 + (-8.45 - 14.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (273. + 473. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 189.T + 7.04e5T^{2} \)
97 \( 1 + (-39.9 - 69.1i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.37419409337882168942743386165, −12.62197630880665013310804724431, −11.91744677060978570694270226077, −10.16703793055303971930942345265, −9.180458980261265280639033310451, −8.708438368557484795751293479696, −6.93215780039022816450789281438, −4.35953560969051629412041860568, −2.30332381371017785866442933859, −1.11941459183221144933306558225, 3.02960595810379522019301643377, 5.54529426834828912101948162998, 6.67925055200820660522989339609, 7.928664245986456847020144844826, 9.139207752352801501022491211991, 9.902709343950467183542345771899, 11.07114904392687547300159586110, 13.59872628511199363578484146654, 14.29795180552881863364118476042, 15.20758712789821723000041737851

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