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Properties

Label 2-37-37.36-c7-0-19
Degree $2$
Conductor $37$
Sign $-0.0970 - 0.995i$
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  − 21.7i·2-s + 44.4·3-s − 345.·4-s − 104. i·5-s − 967. i·6-s − 454.·7-s + 4.72e3i·8-s − 207.·9-s − 2.28e3·10-s − 4.88e3·11-s − 1.53e4·12-s + 2.16e3i·13-s + 9.89e3i·14-s − 4.66e3i·15-s + 5.86e4·16-s − 3.32e4i·17-s + ⋯
L(s)  = 1  − 1.92i·2-s + 0.951·3-s − 2.69·4-s − 0.375i·5-s − 1.82i·6-s − 0.501·7-s + 3.26i·8-s − 0.0947·9-s − 0.721·10-s − 1.10·11-s − 2.56·12-s + 0.273i·13-s + 0.964i·14-s − 0.357i·15-s + 3.57·16-s − 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0970 - 0.995i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0970 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.0970 - 0.995i$
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: \((2,\ 37,\ (\ :7/2),\ -0.0970 - 0.995i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.562672 + 0.620200i\)
\(L(\frac12)\) \(\approx\) \(0.562672 + 0.620200i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.98e4 + 3.06e5i)T \)
good2 \( 1 + 21.7iT - 128T^{2} \)
3 \( 1 - 44.4T + 2.18e3T^{2} \)
5 \( 1 + 104. iT - 7.81e4T^{2} \)
7 \( 1 + 454.T + 8.23e5T^{2} \)
11 \( 1 + 4.88e3T + 1.94e7T^{2} \)
13 \( 1 - 2.16e3iT - 6.27e7T^{2} \)
17 \( 1 + 3.32e4iT - 4.10e8T^{2} \)
19 \( 1 - 2.45e4iT - 8.93e8T^{2} \)
23 \( 1 + 6.23e4iT - 3.40e9T^{2} \)
29 \( 1 + 6.11e4iT - 1.72e10T^{2} \)
31 \( 1 + 3.47e4iT - 2.75e10T^{2} \)
41 \( 1 + 6.45e5T + 1.94e11T^{2} \)
43 \( 1 - 5.24e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.03e6T + 5.06e11T^{2} \)
53 \( 1 - 2.01e6T + 1.17e12T^{2} \)
59 \( 1 + 2.51e6iT - 2.48e12T^{2} \)
61 \( 1 - 9.81e5iT - 3.14e12T^{2} \)
67 \( 1 - 9.41e5T + 6.06e12T^{2} \)
71 \( 1 + 2.67e6T + 9.09e12T^{2} \)
73 \( 1 - 4.46e6T + 1.10e13T^{2} \)
79 \( 1 + 4.48e6iT - 1.92e13T^{2} \)
83 \( 1 - 6.80e6T + 2.71e13T^{2} \)
89 \( 1 + 7.46e5iT - 4.42e13T^{2} \)
97 \( 1 + 3.11e6iT - 8.07e13T^{2} \)
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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

−13.56107128943276360968155803752, −12.74856185186336931782679860444, −11.54957411080722958915923783096, −10.18244963292313006487391376465, −9.221120685690701038587504999800, −8.176733730298646662784106165193, −4.94923475439064634629466285335, −3.29279628708770310615429167286, −2.29945900302183669457931221076, −0.32363272696509855186877692738, 3.40258092358853113645799915170, 5.32938763663962717975835372343, 6.74194696003570712206751812727, 7.985836181740105288603621843466, 8.808333108691584334462976538021, 10.19414390980646658360860794134, 13.01946414740029874417937010250, 13.67898415254394244741242717788, 15.00194030714111491514711994225, 15.31326710402355625342668018362

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