Properties

Label 2-37-37.4-c3-0-0
Degree $2$
Conductor $37$
Sign $-0.988 - 0.148i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
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  + (−2.35 − 0.415i)2-s + (1.60 + 9.11i)3-s + (−2.12 − 0.774i)4-s + (−12.3 − 14.7i)5-s − 22.1i·6-s + (−11.9 + 10.0i)7-s + (21.2 + 12.2i)8-s + (−55.0 + 20.0i)9-s + (23.0 + 40.0i)10-s + (0.249 − 0.432i)11-s + (3.63 − 20.6i)12-s + (−14.0 + 38.5i)13-s + (32.2 − 18.6i)14-s + (114. − 136. i)15-s + (−31.2 − 26.1i)16-s + (22.4 + 61.7i)17-s + ⋯
L(s)  = 1  + (−0.833 − 0.147i)2-s + (0.309 + 1.75i)3-s + (−0.265 − 0.0968i)4-s + (−1.10 − 1.32i)5-s − 1.50i·6-s + (−0.643 + 0.540i)7-s + (0.940 + 0.543i)8-s + (−2.03 + 0.742i)9-s + (0.730 + 1.26i)10-s + (0.00683 − 0.0118i)11-s + (0.0875 − 0.496i)12-s + (−0.299 + 0.822i)13-s + (0.616 − 0.355i)14-s + (1.97 − 2.35i)15-s + (−0.487 − 0.409i)16-s + (0.320 + 0.881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.988 - 0.148i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0225902 + 0.302660i\)
\(L(\frac12)\) \(\approx\) \(0.0225902 + 0.302660i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (99.0 + 202. i)T \)
good2 \( 1 + (2.35 + 0.415i)T + (7.51 + 2.73i)T^{2} \)
3 \( 1 + (-1.60 - 9.11i)T + (-25.3 + 9.23i)T^{2} \)
5 \( 1 + (12.3 + 14.7i)T + (-21.7 + 123. i)T^{2} \)
7 \( 1 + (11.9 - 10.0i)T + (59.5 - 337. i)T^{2} \)
11 \( 1 + (-0.249 + 0.432i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (14.0 - 38.5i)T + (-1.68e3 - 1.41e3i)T^{2} \)
17 \( 1 + (-22.4 - 61.7i)T + (-3.76e3 + 3.15e3i)T^{2} \)
19 \( 1 + (59.0 - 10.4i)T + (6.44e3 - 2.34e3i)T^{2} \)
23 \( 1 + (-86.3 + 49.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-44.6 - 25.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 10.2iT - 2.97e4T^{2} \)
41 \( 1 + (301. + 109. i)T + (5.27e4 + 4.43e4i)T^{2} \)
43 \( 1 - 446. iT - 7.95e4T^{2} \)
47 \( 1 + (32.1 + 55.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (7.64 + 6.41i)T + (2.58e4 + 1.46e5i)T^{2} \)
59 \( 1 + (-541. + 645. i)T + (-3.56e4 - 2.02e5i)T^{2} \)
61 \( 1 + (52.3 - 143. i)T + (-1.73e5 - 1.45e5i)T^{2} \)
67 \( 1 + (6.11 - 5.13i)T + (5.22e4 - 2.96e5i)T^{2} \)
71 \( 1 + (-93.1 - 528. i)T + (-3.36e5 + 1.22e5i)T^{2} \)
73 \( 1 - 69.3T + 3.89e5T^{2} \)
79 \( 1 + (-114. - 136. i)T + (-8.56e4 + 4.85e5i)T^{2} \)
83 \( 1 + (1.06e3 - 388. i)T + (4.38e5 - 3.67e5i)T^{2} \)
89 \( 1 + (796. - 949. i)T + (-1.22e5 - 6.94e5i)T^{2} \)
97 \( 1 + (-191. + 110. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−16.48710323421726100158385157864, −15.62723555076957870965648832669, −14.53303854749596423324607501845, −12.67970663927780905274533251284, −11.19505441131824727104778281773, −9.893000759932329406968615955032, −8.942924136948924666882018708970, −8.378079562888686854206933513154, −5.03066254657566467268623746338, −3.99948950919026252656925703941, 0.33035455126225517760810904395, 3.18234626846795769204840970033, 6.88058665766874017642238415440, 7.36735696427433296734361050320, 8.415032288837722119257485242853, 10.29862505007087564696572753833, 11.74916180021092304078807839014, 13.01508272905665294703994534191, 13.95034680824796417260683923211, 15.28188041813465004272639922544

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