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Properties

Label 2-37-37.36-c7-0-18
Degree $2$
Conductor $37$
Sign $0.0639 - 0.997i$
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  − 7.52i·2-s − 79.9·3-s + 71.3·4-s − 468. i·5-s + 601. i·6-s − 1.53e3·7-s − 1.50e3i·8-s + 4.20e3·9-s − 3.52e3·10-s + 1.24e3·11-s − 5.70e3·12-s + 9.01e3i·13-s + 1.15e4i·14-s + 3.74e4i·15-s − 2.15e3·16-s − 1.05e4i·17-s + ⋯
L(s)  = 1  − 0.665i·2-s − 1.70·3-s + 0.557·4-s − 1.67i·5-s + 1.13i·6-s − 1.69·7-s − 1.03i·8-s + 1.92·9-s − 1.11·10-s + 0.281·11-s − 0.953·12-s + 1.13i·13-s + 1.12i·14-s + 2.86i·15-s − 0.131·16-s − 0.520i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.0639 - 0.997i$
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: \((2,\ 37,\ (\ :7/2),\ 0.0639 - 0.997i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0618540 + 0.0580194i\)
\(L(\frac12)\) \(\approx\) \(0.0618540 + 0.0580194i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.96e4 + 3.07e5i)T \)
good2 \( 1 + 7.52iT - 128T^{2} \)
3 \( 1 + 79.9T + 2.18e3T^{2} \)
5 \( 1 + 468. iT - 7.81e4T^{2} \)
7 \( 1 + 1.53e3T + 8.23e5T^{2} \)
11 \( 1 - 1.24e3T + 1.94e7T^{2} \)
13 \( 1 - 9.01e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.05e4iT - 4.10e8T^{2} \)
19 \( 1 - 4.66e4iT - 8.93e8T^{2} \)
23 \( 1 + 3.20e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.96e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.02e5iT - 2.75e10T^{2} \)
41 \( 1 + 6.47e5T + 1.94e11T^{2} \)
43 \( 1 - 3.00e5iT - 2.71e11T^{2} \)
47 \( 1 - 3.72e5T + 5.06e11T^{2} \)
53 \( 1 - 3.08e5T + 1.17e12T^{2} \)
59 \( 1 - 4.70e5iT - 2.48e12T^{2} \)
61 \( 1 - 9.16e5iT - 3.14e12T^{2} \)
67 \( 1 + 2.61e6T + 6.06e12T^{2} \)
71 \( 1 + 3.34e6T + 9.09e12T^{2} \)
73 \( 1 - 7.25e5T + 1.10e13T^{2} \)
79 \( 1 - 1.65e5iT - 1.92e13T^{2} \)
83 \( 1 + 5.82e6T + 2.71e13T^{2} \)
89 \( 1 - 4.14e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.09e6iT - 8.07e13T^{2} \)
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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

−13.15986252071915308721400180114, −12.22722907488175105510731161985, −11.94348236703736977911421951387, −10.32759254167086758542218374740, −9.354474070969619386963966148627, −6.79571095410369697921947090212, −5.78021876099778837468397205189, −4.11793901775873027120946254989, −1.31431081985530182717620872660, −0.04858083825840207998894761084, 3.05144749080141820137299683725, 5.73332684666274797075399554826, 6.55970520480040024102196815654, 7.13425510745545462217499806180, 10.10399170803161053672804339567, 10.85110783676496683752264541285, 11.85515283209425536745491608997, 13.27196310978814898368570456766, 15.23338585781846264099815716180, 15.65808152596701113065991148832

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