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Properties

Label 2-37-37.6-c8-0-17
Degree $2$
Conductor $37$
Sign $0.347 + 0.937i$
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  + (17.8 − 17.8i)2-s + 43.1i·3-s − 377. i·4-s + (137. + 137. i)5-s + (767. + 767. i)6-s + 2.62e3·7-s + (−2.17e3 − 2.17e3i)8-s + 4.70e3·9-s + 4.88e3·10-s − 1.50e4i·11-s + 1.62e4·12-s + (−2.67e4 − 2.67e4i)13-s + (4.67e4 − 4.67e4i)14-s + (−5.91e3 + 5.91e3i)15-s + 1.94e4·16-s + (5.59e4 + 5.59e4i)17-s + ⋯
L(s)  = 1  + (1.11 − 1.11i)2-s + 0.532i·3-s − 1.47i·4-s + (0.219 + 0.219i)5-s + (0.592 + 0.592i)6-s + 1.09·7-s + (−0.530 − 0.530i)8-s + 0.716·9-s + 0.488·10-s − 1.02i·11-s + 0.785·12-s + (−0.937 − 0.937i)13-s + (1.21 − 1.21i)14-s + (−0.116 + 0.116i)15-s + 0.296·16-s + (0.669 + 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.347 + 0.937i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ 0.347 + 0.937i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.20442 - 2.22952i\)
\(L(\frac12)\) \(\approx\) \(3.20442 - 2.22952i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (6.38e5 - 1.76e6i)T \)
good2 \( 1 + (-17.8 + 17.8i)T - 256iT^{2} \)
3 \( 1 - 43.1iT - 6.56e3T^{2} \)
5 \( 1 + (-137. - 137. i)T + 3.90e5iT^{2} \)
7 \( 1 - 2.62e3T + 5.76e6T^{2} \)
11 \( 1 + 1.50e4iT - 2.14e8T^{2} \)
13 \( 1 + (2.67e4 + 2.67e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-5.59e4 - 5.59e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-4.50e4 - 4.50e4i)T + 1.69e10iT^{2} \)
23 \( 1 + (7.84e4 + 7.84e4i)T + 7.83e10iT^{2} \)
29 \( 1 + (5.69e5 - 5.69e5i)T - 5.00e11iT^{2} \)
31 \( 1 + (-1.81e5 + 1.81e5i)T - 8.52e11iT^{2} \)
41 \( 1 - 8.32e5iT - 7.98e12T^{2} \)
43 \( 1 + (-2.76e6 - 2.76e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 6.19e6T + 2.38e13T^{2} \)
53 \( 1 - 1.20e6T + 6.22e13T^{2} \)
59 \( 1 + (-5.72e6 - 5.72e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.26e7 - 1.26e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 3.36e6iT - 4.06e14T^{2} \)
71 \( 1 + 3.21e7T + 6.45e14T^{2} \)
73 \( 1 + 4.86e6iT - 8.06e14T^{2} \)
79 \( 1 + (3.49e7 + 3.49e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 5.87e6T + 2.25e15T^{2} \)
89 \( 1 + (4.68e7 - 4.68e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (2.39e7 + 2.39e7i)T + 7.83e15iT^{2} \)
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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

−14.33099705022334174146256826347, −13.06824044614939873271581635581, −11.97079630758347341081768215475, −10.79000446475502252116425746856, −10.05005728418930298411132260151, −7.979539446622958495376111741024, −5.60993809163255624362585929280, −4.49023504806382437031689141354, −3.09370141501972735887850661195, −1.42576719929706569403793445773, 1.76374909572051006497535294051, 4.34960171155119951086873291043, 5.30050958107502336226899298355, 7.06501027685036129788234616944, 7.63124676885773555641069695599, 9.653556710381873148278412375401, 11.78694199256541670501488808035, 12.74433815152080468875240490264, 13.88163546871749224164708777031, 14.68162044215596123456350506736

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