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Properties

Label 2-63-63.58-c3-0-16
Degree $2$
Conductor $63$
Sign $0.927 + 0.372i$
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  + 4.21·2-s + (2.84 − 4.34i)3-s + 9.76·4-s + (3.91 + 6.77i)5-s + (11.9 − 18.3i)6-s + (−15.9 + 9.49i)7-s + 7.43·8-s + (−10.8 − 24.7i)9-s + (16.4 + 28.5i)10-s + (6.42 − 11.1i)11-s + (27.7 − 42.4i)12-s + (6.74 − 11.6i)13-s + (−67.0 + 40.0i)14-s + (40.5 + 2.25i)15-s − 46.7·16-s + (35.5 + 61.4i)17-s + ⋯
L(s)  = 1  + 1.49·2-s + (0.547 − 0.836i)3-s + 1.22·4-s + (0.349 + 0.606i)5-s + (0.815 − 1.24i)6-s + (−0.858 + 0.512i)7-s + 0.328·8-s + (−0.400 − 0.916i)9-s + (0.521 + 0.903i)10-s + (0.176 − 0.304i)11-s + (0.668 − 1.02i)12-s + (0.143 − 0.249i)13-s + (−1.27 + 0.764i)14-s + (0.698 + 0.0388i)15-s − 0.730·16-s + (0.506 + 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.927 + 0.372i$
Analytic conductor: \(3.71712\)
Root analytic conductor: \(1.92798\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 63,\ (\ :3/2),\ 0.927 + 0.372i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.14261 - 0.607800i\)
\(L(\frac12)\) \(\approx\) \(3.14261 - 0.607800i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.84 + 4.34i)T \)
7 \( 1 + (15.9 - 9.49i)T \)
good2 \( 1 - 4.21T + 8T^{2} \)
5 \( 1 + (-3.91 - 6.77i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.42 + 11.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-6.74 + 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-35.5 - 61.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (46.2 - 80.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (1.97 + 3.41i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (90.3 + 156. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 270.T + 2.97e4T^{2} \)
37 \( 1 + (-110. + 190. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (33.6 - 58.2i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-237. - 411. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 512.T + 1.03e5T^{2} \)
53 \( 1 + (238. + 413. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 717.T + 2.05e5T^{2} \)
61 \( 1 - 376.T + 2.26e5T^{2} \)
67 \( 1 - 694.T + 3.00e5T^{2} \)
71 \( 1 + 230.T + 3.57e5T^{2} \)
73 \( 1 + (258. + 446. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 943.T + 4.93e5T^{2} \)
83 \( 1 + (84.4 + 146. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (149. - 258. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-389. - 674. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.37821514122363545937873108268, −13.19669289671581341878090767838, −12.66354830099479144057766194851, −11.58161242374596861864444388571, −9.856079651526114888803229665042, −8.243996723834061199857104051764, −6.46884093616515603711306444911, −5.98078593707582746896907286458, −3.67890502423644957319564951329, −2.51220036280650991821873675818, 2.93740111628088873273683722121, 4.25683433305785743198815011776, 5.30152034350573383825396231619, 6.87821592561891542320569079992, 8.909853814707819399128726514777, 9.915156986205390225633696358657, 11.38338317313020078261945010597, 12.76976771539271477902367050104, 13.51223075075233620657704804588, 14.32419510833773584859016086238

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