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Properties

Label 2-37-37.26-c7-0-15
Degree $2$
Conductor $37$
Sign $0.823 - 0.567i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
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  + (9.33 + 16.1i)2-s + (28.0 − 48.6i)3-s + (−110. + 190. i)4-s + (166. − 288. i)5-s + 1.04e3·6-s + (1.55 − 2.69i)7-s − 1.72e3·8-s + (−483. − 837. i)9-s + 6.21e3·10-s + 7.98e3·11-s + (6.18e3 + 1.07e4i)12-s + (586. − 1.01e3i)13-s + 58.0·14-s + (−9.35e3 − 1.62e4i)15-s + (−1.99e3 − 3.44e3i)16-s + (−7.87e3 − 1.36e4i)17-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)2-s + (0.600 − 1.04i)3-s + (−0.860 + 1.49i)4-s + (0.595 − 1.03i)5-s + 1.98·6-s + (0.00171 − 0.00296i)7-s − 1.19·8-s + (−0.221 − 0.383i)9-s + 1.96·10-s + 1.80·11-s + (1.03 + 1.79i)12-s + (0.0740 − 0.128i)13-s + 0.00565·14-s + (−0.715 − 1.23i)15-s + (−0.121 − 0.210i)16-s + (−0.388 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.43880 + 1.07039i\)
\(L(\frac12)\) \(\approx\) \(3.43880 + 1.07039i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (6.93e4 + 3.00e5i)T \)
good2 \( 1 + (-9.33 - 16.1i)T + (-64 + 110. i)T^{2} \)
3 \( 1 + (-28.0 + 48.6i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-166. + 288. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-1.55 + 2.69i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 - 7.98e3T + 1.94e7T^{2} \)
13 \( 1 + (-586. + 1.01e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (7.87e3 + 1.36e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (9.47e3 - 1.64e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 8.39e4T + 3.40e9T^{2} \)
29 \( 1 + 1.47e5T + 1.72e10T^{2} \)
31 \( 1 - 6.72e4T + 2.75e10T^{2} \)
41 \( 1 + (2.22e5 - 3.84e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 - 2.25e5T + 2.71e11T^{2} \)
47 \( 1 - 6.08e4T + 5.06e11T^{2} \)
53 \( 1 + (8.96e5 + 1.55e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-2.52e5 - 4.38e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.35e6 - 2.34e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.47e6 - 2.56e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (1.69e5 - 2.92e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 3.64e6T + 1.10e13T^{2} \)
79 \( 1 + (3.04e6 - 5.28e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (3.48e6 + 6.03e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (-5.25e6 - 9.10e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 8.11e6T + 8.07e13T^{2} \)
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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

−14.59787742031062354323132031063, −13.89513273112084831470531563326, −13.06535067666994831436345301877, −12.11400465647707140066294847911, −9.231350120162398718143407558312, −8.181785534174200042094816846666, −6.92653271686622910813682366298, −5.82071297396743413538013248386, −4.19429237099007246575511540392, −1.54951574024820766190622008614, 1.92492932681600561254918796115, 3.40380530829816353877198705595, 4.28063223320455903661980790417, 6.34375564779585340409034650935, 9.131972781545276391154535776692, 10.05417397421815524503088670647, 10.96681850580895829891055018005, 12.14188047461814883726506749815, 13.78277363136178183620238201936, 14.39220659984488735483226553558

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