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Properties

Label 2-37-37.26-c3-0-3
Degree $2$
Conductor $37$
Sign $-0.615 - 0.788i$
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  + (2.25 + 3.90i)2-s + (−3.09 + 5.36i)3-s + (−6.17 + 10.6i)4-s + (10.3 − 17.9i)5-s − 27.9·6-s + (−7.44 + 12.8i)7-s − 19.6·8-s + (−5.65 − 9.79i)9-s + 93.3·10-s + 27.5·11-s + (−38.2 − 66.2i)12-s + (−11.6 + 20.2i)13-s − 67.1·14-s + (64.0 + 110. i)15-s + (5.13 + 8.89i)16-s + (−58.6 − 101. i)17-s + ⋯
L(s)  = 1  + (0.797 + 1.38i)2-s + (−0.595 + 1.03i)3-s + (−0.771 + 1.33i)4-s + (0.925 − 1.60i)5-s − 1.89·6-s + (−0.402 + 0.696i)7-s − 0.867·8-s + (−0.209 − 0.362i)9-s + 2.95·10-s + 0.755·11-s + (−0.919 − 1.59i)12-s + (−0.249 + 0.431i)13-s − 1.28·14-s + (1.10 + 1.90i)15-s + (0.0802 + 0.138i)16-s + (−0.836 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.737552 + 1.51065i\)
\(L(\frac12)\) \(\approx\) \(0.737552 + 1.51065i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (215. - 63.4i)T \)
good2 \( 1 + (-2.25 - 3.90i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (3.09 - 5.36i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-10.3 + 17.9i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (7.44 - 12.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 27.5T + 1.33e3T^{2} \)
13 \( 1 + (11.6 - 20.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (58.6 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-47.0 + 81.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 36.3T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 + 62.4T + 2.97e4T^{2} \)
41 \( 1 + (75.6 - 131. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 99.0T + 7.95e4T^{2} \)
47 \( 1 + 382.T + 1.03e5T^{2} \)
53 \( 1 + (37.9 + 65.7i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (47.5 + 82.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-217. + 377. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (54.8 - 94.9i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (101. - 176. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 23.1T + 3.89e5T^{2} \)
79 \( 1 + (227. - 393. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-283. - 490. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-50.8 - 88.0i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 451.T + 9.12e5T^{2} \)
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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

−16.09505209363243422734211035740, −15.64189933187694214176994953432, −14.02632890746567088377224063712, −13.11240513023600364552124924088, −11.81398616510391008209497929673, −9.607054108041413939772570711231, −8.829035184406321879040516153810, −6.55300784797309969229025264538, −5.17787274462318295998913313991, −4.70438958560095807754062368234, 1.73715991613505136488708017569, 3.48741479438414879559890507933, 6.03235930045668838197129860192, 7.02036590522099107580480201892, 10.04581912751697467627340504218, 10.72264690123505657938824524022, 11.90009652124726472964890000178, 13.01134522068322509268075556062, 13.83299352161875797672374510156, 14.76299505143247707439709411824

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