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Properties

Label 2-37-37.31-c8-0-6
Degree $2$
Conductor $37$
Sign $0.999 + 0.0116i$
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  + (−15.5 − 15.5i)2-s + 38.2i·3-s + 228. i·4-s + (431. − 431. i)5-s + (595. − 595. i)6-s − 3.11e3·7-s + (−426. + 426. i)8-s + 5.09e3·9-s − 1.34e4·10-s + 6.13e3i·11-s − 8.74e3·12-s + (−3.39e4 + 3.39e4i)13-s + (4.84e4 + 4.84e4i)14-s + (1.65e4 + 1.65e4i)15-s + 7.17e4·16-s + (4.44e4 − 4.44e4i)17-s + ⋯
L(s)  = 1  + (−0.972 − 0.972i)2-s + 0.472i·3-s + 0.892i·4-s + (0.690 − 0.690i)5-s + (0.459 − 0.459i)6-s − 1.29·7-s + (−0.104 + 0.104i)8-s + 0.776·9-s − 1.34·10-s + 0.419i·11-s − 0.421·12-s + (−1.18 + 1.18i)13-s + (1.26 + 1.26i)14-s + (0.326 + 0.326i)15-s + 1.09·16-s + (0.532 − 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.999 + 0.0116i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ 0.999 + 0.0116i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.908101 - 0.00527993i\)
\(L(\frac12)\) \(\approx\) \(0.908101 - 0.00527993i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.44e6 + 1.19e6i)T \)
good2 \( 1 + (15.5 + 15.5i)T + 256iT^{2} \)
3 \( 1 - 38.2iT - 6.56e3T^{2} \)
5 \( 1 + (-431. + 431. i)T - 3.90e5iT^{2} \)
7 \( 1 + 3.11e3T + 5.76e6T^{2} \)
11 \( 1 - 6.13e3iT - 2.14e8T^{2} \)
13 \( 1 + (3.39e4 - 3.39e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-4.44e4 + 4.44e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (4.72e3 - 4.72e3i)T - 1.69e10iT^{2} \)
23 \( 1 + (-3.42e5 + 3.42e5i)T - 7.83e10iT^{2} \)
29 \( 1 + (-9.31e5 - 9.31e5i)T + 5.00e11iT^{2} \)
31 \( 1 + (-5.45e5 - 5.45e5i)T + 8.52e11iT^{2} \)
41 \( 1 - 4.15e6iT - 7.98e12T^{2} \)
43 \( 1 + (-1.63e6 + 1.63e6i)T - 1.16e13iT^{2} \)
47 \( 1 + 4.24e6T + 2.38e13T^{2} \)
53 \( 1 - 6.45e6T + 6.22e13T^{2} \)
59 \( 1 + (1.25e7 - 1.25e7i)T - 1.46e14iT^{2} \)
61 \( 1 + (-1.05e7 - 1.05e7i)T + 1.91e14iT^{2} \)
67 \( 1 + 2.69e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.50e7T + 6.45e14T^{2} \)
73 \( 1 - 3.15e7iT - 8.06e14T^{2} \)
79 \( 1 + (1.31e7 - 1.31e7i)T - 1.51e15iT^{2} \)
83 \( 1 - 2.74e7T + 2.25e15T^{2} \)
89 \( 1 + (2.94e7 + 2.94e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (3.41e7 - 3.41e7i)T - 7.83e15iT^{2} \)
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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

−14.61407264925738115321875657299, −12.91362504533313476752080135932, −12.15031156512771768703814887286, −10.39757328107409307357123112191, −9.624660427510132298154362117167, −9.064339844789237162789539673013, −6.86807216435565354848772448608, −4.80922042951850210493355827632, −2.75209400169455469454710677925, −1.13401769720705565906677698213, 0.60475443082330414503411118271, 2.95109845835303650681173670623, 5.94580816976440044903041150335, 6.85643373510288421892006736165, 7.896866761641043571484390640846, 9.696835718618156575059409270515, 10.14044826310926856112307228783, 12.45857951866685516790845807068, 13.43371709142071291262499542507, 15.02675772549177916341096271043

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