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Properties

Label 2-37-37.6-c2-0-1
Degree $2$
Conductor $37$
Sign $0.343 - 0.939i$
Analytic cond. $1.00817$
Root an. cond. $1.00408$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 + 1.04i)2-s − 0.377i·3-s + 1.79i·4-s + (5.26 + 5.26i)5-s + (0.396 + 0.396i)6-s − 5.54·7-s + (−6.08 − 6.08i)8-s + 8.85·9-s − 11.0·10-s − 18.1i·11-s + 0.678·12-s + (7.75 + 7.75i)13-s + (5.81 − 5.81i)14-s + (1.99 − 1.99i)15-s + 5.58·16-s + (−18.1 − 18.1i)17-s + ⋯
L(s)  = 1  + (−0.524 + 0.524i)2-s − 0.125i·3-s + 0.449i·4-s + (1.05 + 1.05i)5-s + (0.0661 + 0.0661i)6-s − 0.791·7-s + (−0.760 − 0.760i)8-s + 0.984·9-s − 1.10·10-s − 1.65i·11-s + 0.0565·12-s + (0.596 + 0.596i)13-s + (0.415 − 0.415i)14-s + (0.132 − 0.132i)15-s + 0.349·16-s + (−1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.343 - 0.939i$
Analytic conductor: \(1.00817\)
Root analytic conductor: \(1.00408\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1),\ 0.343 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.731337 + 0.511336i\)
\(L(\frac12)\) \(\approx\) \(0.731337 + 0.511336i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (32.1 - 18.3i)T \)
good2 \( 1 + (1.04 - 1.04i)T - 4iT^{2} \)
3 \( 1 + 0.377iT - 9T^{2} \)
5 \( 1 + (-5.26 - 5.26i)T + 25iT^{2} \)
7 \( 1 + 5.54T + 49T^{2} \)
11 \( 1 + 18.1iT - 121T^{2} \)
13 \( 1 + (-7.75 - 7.75i)T + 169iT^{2} \)
17 \( 1 + (18.1 + 18.1i)T + 289iT^{2} \)
19 \( 1 + (-2.37 - 2.37i)T + 361iT^{2} \)
23 \( 1 + (-16.3 - 16.3i)T + 529iT^{2} \)
29 \( 1 + (-20.1 + 20.1i)T - 841iT^{2} \)
31 \( 1 + (10.4 - 10.4i)T - 961iT^{2} \)
41 \( 1 - 2.51iT - 1.68e3T^{2} \)
43 \( 1 + (19.0 + 19.0i)T + 1.84e3iT^{2} \)
47 \( 1 + 33.0T + 2.20e3T^{2} \)
53 \( 1 + 39.9T + 2.80e3T^{2} \)
59 \( 1 + (-46.2 - 46.2i)T + 3.48e3iT^{2} \)
61 \( 1 + (-3.22 + 3.22i)T - 3.72e3iT^{2} \)
67 \( 1 + 44.2iT - 4.48e3T^{2} \)
71 \( 1 + 92.4T + 5.04e3T^{2} \)
73 \( 1 - 87.3iT - 5.32e3T^{2} \)
79 \( 1 + (-39.7 - 39.7i)T + 6.24e3iT^{2} \)
83 \( 1 + 112.T + 6.88e3T^{2} \)
89 \( 1 + (-113. + 113. i)T - 7.92e3iT^{2} \)
97 \( 1 + (6.42 + 6.42i)T + 9.40e3iT^{2} \)
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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

−16.33861450157295022409219602610, −15.64347559688776490930309787211, −13.81520790260981081921312370491, −13.21263409957079378464616121891, −11.37293764508835621832235524453, −9.936097127202211573880678579203, −8.853147888446017423064871190970, −7.00381686294681255676011559608, −6.28772541155145700474865303869, −3.20224590492805940480038951598, 1.71556156900858901497764893443, 4.86883614315670739913294515655, 6.49751485043993022481869572067, 8.858990155168010760507635587815, 9.768675720676908560243771443196, 10.51716699070245421369582662430, 12.60996470691691341172630620577, 13.11619608342247146771917981273, 14.91439841806865854375295835677, 15.97782132901047869854339261059

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