Properties

Label 2-38-19.7-c7-0-10
Degree $2$
Conductor $38$
Sign $0.998 - 0.0463i$
Analytic cond. $11.8706$
Root an. cond. $3.44537$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 6.92i)2-s + (7.15 + 12.3i)3-s + (−31.9 + 55.4i)4-s + (−222. − 385. i)5-s + (−57.2 + 99.1i)6-s + 1.43e3·7-s − 511.·8-s + (991. − 1.71e3i)9-s + (1.78e3 − 3.08e3i)10-s + 5.16e3·11-s − 915.·12-s + (−4.98e3 + 8.63e3i)13-s + (5.74e3 + 9.94e3i)14-s + (3.18e3 − 5.51e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (−5.59e3 − 9.68e3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.153 + 0.265i)3-s + (−0.249 + 0.433i)4-s + (−0.796 − 1.37i)5-s + (−0.108 + 0.187i)6-s + 1.58·7-s − 0.353·8-s + (0.453 − 0.784i)9-s + (0.563 − 0.975i)10-s + 1.16·11-s − 0.153·12-s + (−0.629 + 1.09i)13-s + (0.559 + 0.969i)14-s + (0.243 − 0.422i)15-s + (−0.125 − 0.216i)16-s + (−0.276 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0463i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38\)    =    \(2 \cdot 19\)
Sign: $0.998 - 0.0463i$
Analytic conductor: \(11.8706\)
Root analytic conductor: \(3.44537\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{38} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 38,\ (\ :7/2),\ 0.998 - 0.0463i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.30939 + 0.0535568i\)
\(L(\frac12)\) \(\approx\) \(2.30939 + 0.0535568i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 6.92i)T \)
19 \( 1 + (-2.88e4 + 7.66e3i)T \)
good3 \( 1 + (-7.15 - 12.3i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (222. + 385. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 - 1.43e3T + 8.23e5T^{2} \)
11 \( 1 - 5.16e3T + 1.94e7T^{2} \)
13 \( 1 + (4.98e3 - 8.63e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (5.59e3 + 9.68e3i)T + (-2.05e8 + 3.55e8i)T^{2} \)
23 \( 1 + (-3.34e4 + 5.78e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-5.89e4 + 1.02e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 - 1.54e5T + 2.75e10T^{2} \)
37 \( 1 + 4.78e5T + 9.49e10T^{2} \)
41 \( 1 + (1.78e5 + 3.08e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (1.75e4 + 3.03e4i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (5.74e5 - 9.94e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-1.74e5 + 3.01e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-5.09e5 - 8.82e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (9.24e5 - 1.60e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.34e6 - 2.33e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-1.51e6 - 2.62e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + (2.86e6 + 4.96e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-2.37e6 - 4.10e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 2.86e6T + 2.71e13T^{2} \)
89 \( 1 + (5.58e6 - 9.67e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (3.49e6 + 6.05e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−14.83897102939003318596545102770, −13.90754449186255122759912747996, −12.12457628727027159311972330840, −11.73642139701582679002574832812, −9.252897501573561307868907830813, −8.446760676377081977812201499614, −7.03316127475864092848670789230, −4.82817869085080340466794061981, −4.24374740158509935866302506001, −1.09782176147166385029670770638, 1.60327645031020513536435969366, 3.31185577405350268675449380372, 4.93732910183986903532238306897, 7.10187611796924205446916670932, 8.147816757155324563552896162836, 10.29571580734208187167682834243, 11.21891135016877806737198536934, 12.04750501619627718752203205427, 13.79300565587211002055503399576, 14.62938653020005790125379269148

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