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Properties

Label 2-37-37.36-c1-0-1
Degree $2$
Conductor $37$
Sign $0.164 + 0.986i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 3-s − 2·4-s + 2i·5-s + 2i·6-s + 3·7-s − 2·9-s + 4·10-s − 3·11-s + 2·12-s + 6i·13-s − 6i·14-s − 2i·15-s − 4·16-s − 2i·17-s + 4i·18-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.577·3-s − 4-s + 0.894i·5-s + 0.816i·6-s + 1.13·7-s − 0.666·9-s + 1.26·10-s − 0.904·11-s + 0.577·12-s + 1.66i·13-s − 1.60i·14-s − 0.516i·15-s − 16-s − 0.485i·17-s + 0.942i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510028 - 0.432059i\)
\(L(\frac12)\) \(\approx\) \(0.510028 - 0.432059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1 + 6i)T \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 - 2iT - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−16.35243031038882138361455797987, −14.69261999774586902257055533599, −13.66233242554910635758515229827, −12.03641000418996993895043545230, −11.17371330004469292285283552681, −10.70164083949079417779568466719, −8.962523271675309961068729949697, −6.91264800434440678178571984743, −4.76215737775444926855823515210, −2.55062267981930742747221241338, 5.12350748025181051471923975855, 5.72792428408809935011656816428, 7.87661949667809268185967698094, 8.405568605735233663479975827973, 10.57595062939260961208204296645, 12.00512485502068726851633512346, 13.41483824783508981828592608142, 14.74467812005020822162815223724, 15.61340145096855589033951217458, 16.80724976697154758132500055494

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