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Properties

Label 2-37-37.36-c7-0-16
Degree $2$
Conductor $37$
Sign $-0.587 + 0.809i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.3i·2-s + 6.68·3-s + 21.4·4-s − 296. i·5-s + 69.0i·6-s − 1.26e3·7-s + 1.54e3i·8-s − 2.14e3·9-s + 3.05e3·10-s − 5.47e3·11-s + 143.·12-s − 2.48e3i·13-s − 1.30e4i·14-s − 1.98e3i·15-s − 1.31e4·16-s + 2.74e3i·17-s + ⋯
L(s)  = 1  + 0.912i·2-s + 0.143·3-s + 0.167·4-s − 1.05i·5-s + 0.130i·6-s − 1.39·7-s + 1.06i·8-s − 0.979·9-s + 0.966·10-s − 1.23·11-s + 0.0239·12-s − 0.314i·13-s − 1.26i·14-s − 0.151i·15-s − 0.804·16-s + 0.135i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0573670 - 0.112571i\)
\(L(\frac12)\) \(\approx\) \(0.0573670 - 0.112571i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.81e5 - 2.49e5i)T \)
good2 \( 1 - 10.3iT - 128T^{2} \)
3 \( 1 - 6.68T + 2.18e3T^{2} \)
5 \( 1 + 296. iT - 7.81e4T^{2} \)
7 \( 1 + 1.26e3T + 8.23e5T^{2} \)
11 \( 1 + 5.47e3T + 1.94e7T^{2} \)
13 \( 1 + 2.48e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.74e3iT - 4.10e8T^{2} \)
19 \( 1 - 1.00e4iT - 8.93e8T^{2} \)
23 \( 1 + 7.92e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.38e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.22e5iT - 2.75e10T^{2} \)
41 \( 1 - 4.66e5T + 1.94e11T^{2} \)
43 \( 1 + 2.52e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.81e5T + 5.06e11T^{2} \)
53 \( 1 + 1.70e6T + 1.17e12T^{2} \)
59 \( 1 + 2.13e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.76e6iT - 3.14e12T^{2} \)
67 \( 1 - 1.24e5T + 6.06e12T^{2} \)
71 \( 1 + 2.45e6T + 9.09e12T^{2} \)
73 \( 1 + 4.61e6T + 1.10e13T^{2} \)
79 \( 1 + 2.29e6iT - 1.92e13T^{2} \)
83 \( 1 - 8.43e6T + 2.71e13T^{2} \)
89 \( 1 + 2.22e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.54e7iT - 8.07e13T^{2} \)
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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

−14.67669528384396451125730631651, −13.26125577659283810348110322748, −12.32316824877657000207472050780, −10.59387259713001019368865358269, −8.959326214580294661399709806502, −7.944601963532384128137649745845, −6.32780538113648532212479857514, −5.21133188856591109688930437971, −2.80041081897152907062868583088, −0.04890060272000120581103048580, 2.58458486910005267560684361807, 3.30456783879076690965294598926, 6.07735048944718636431463086379, 7.34606355644350213101404694514, 9.441928392857939079020801138816, 10.50794328719956599674211061150, 11.42027826251404895224167588149, 12.75502199340144801683955754284, 13.82834019231728541139665571996, 15.35852167275501181921454455247

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