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Properties

Label 2-37-37.6-c8-0-21
Degree $2$
Conductor $37$
Sign $-0.914 - 0.404i$
Analytic cond. $15.0730$
Root an. cond. $3.88239$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.2 − 19.2i)2-s + 18.4i·3-s − 485. i·4-s + (−521. − 521. i)5-s + (356. + 356. i)6-s − 2.68e3·7-s + (−4.41e3 − 4.41e3i)8-s + 6.21e3·9-s − 2.00e4·10-s + 1.87e4i·11-s + 8.97e3·12-s + (−2.44e4 − 2.44e4i)13-s + (−5.17e4 + 5.17e4i)14-s + (9.63e3 − 9.63e3i)15-s − 4.58e4·16-s + (−6.25e4 − 6.25e4i)17-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + 0.228i·3-s − 1.89i·4-s + (−0.833 − 0.833i)5-s + (0.274 + 0.274i)6-s − 1.11·7-s + (−1.07 − 1.07i)8-s + 0.947·9-s − 2.00·10-s + 1.28i·11-s + 0.432·12-s + (−0.855 − 0.855i)13-s + (−1.34 + 1.34i)14-s + (0.190 − 0.190i)15-s − 0.699·16-s + (−0.749 − 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.378667 + 1.79432i\)
\(L(\frac12)\) \(\approx\) \(0.378667 + 1.79432i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (8.18e5 + 1.68e6i)T \)
good2 \( 1 + (-19.2 + 19.2i)T - 256iT^{2} \)
3 \( 1 - 18.4iT - 6.56e3T^{2} \)
5 \( 1 + (521. + 521. i)T + 3.90e5iT^{2} \)
7 \( 1 + 2.68e3T + 5.76e6T^{2} \)
11 \( 1 - 1.87e4iT - 2.14e8T^{2} \)
13 \( 1 + (2.44e4 + 2.44e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (6.25e4 + 6.25e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (1.65e5 + 1.65e5i)T + 1.69e10iT^{2} \)
23 \( 1 + (-2.90e5 - 2.90e5i)T + 7.83e10iT^{2} \)
29 \( 1 + (-6.30e5 + 6.30e5i)T - 5.00e11iT^{2} \)
31 \( 1 + (-4.57e5 + 4.57e5i)T - 8.52e11iT^{2} \)
41 \( 1 + 4.51e5iT - 7.98e12T^{2} \)
43 \( 1 + (1.92e6 + 1.92e6i)T + 1.16e13iT^{2} \)
47 \( 1 - 5.99e6T + 2.38e13T^{2} \)
53 \( 1 + 1.66e6T + 6.22e13T^{2} \)
59 \( 1 + (-7.81e6 - 7.81e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (2.23e6 - 2.23e6i)T - 1.91e14iT^{2} \)
67 \( 1 - 6.85e6iT - 4.06e14T^{2} \)
71 \( 1 + 5.97e6T + 6.45e14T^{2} \)
73 \( 1 + 4.39e6iT - 8.06e14T^{2} \)
79 \( 1 + (-1.49e7 - 1.49e7i)T + 1.51e15iT^{2} \)
83 \( 1 + 6.15e7T + 2.25e15T^{2} \)
89 \( 1 + (-3.34e7 + 3.34e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-7.90e7 - 7.90e7i)T + 7.83e15iT^{2} \)
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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

−13.23967562871273904868273905586, −12.76537040816004887913534851426, −11.90560768140782720563694915883, −10.40689995805582726327475183707, −9.391046519136827784558007864885, −7.08937038906573032482268563354, −4.89819582743078834534330247721, −4.16109190578715120119001405669, −2.52559042808245631073772728133, −0.48542268543348050269985592651, 3.21892606961462059096037359354, 4.35674200425681173968825725521, 6.45678420548174453540564553100, 6.84796649309974035721872625910, 8.360284904565285362478113919950, 10.53418468505537499409459964863, 12.25164768877604911405508713049, 13.06726770378291065171573477239, 14.29542637643802141168321745284, 15.19761088830133814319490067681

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