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Properties

Label 2-37-37.11-c5-0-13
Degree $2$
Conductor $37$
Sign $0.0536 + 0.998i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.81 − 4.51i)2-s + (10.9 − 18.8i)3-s + (24.7 − 42.8i)4-s + (44.4 + 25.6i)5-s − 196. i·6-s + (−121. + 210. i)7-s − 157. i·8-s + (−116. − 201. i)9-s + 463.·10-s − 232.·11-s + (−539. − 934. i)12-s + (−178. − 102. i)13-s + 2.19e3i·14-s + (969. − 559. i)15-s + (79.3 + 137. i)16-s + (914. − 528. i)17-s + ⋯
L(s)  = 1  + (1.38 − 0.797i)2-s + (0.699 − 1.21i)3-s + (0.773 − 1.33i)4-s + (0.795 + 0.459i)5-s − 2.23i·6-s + (−0.939 + 1.62i)7-s − 0.871i·8-s + (−0.479 − 0.830i)9-s + 1.46·10-s − 0.579·11-s + (−1.08 − 1.87i)12-s + (−0.292 − 0.168i)13-s + 2.99i·14-s + (1.11 − 0.642i)15-s + (0.0775 + 0.134i)16-s + (0.767 − 0.443i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0536 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0536 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.0536 + 0.998i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.0536 + 0.998i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.74763 - 2.60392i\)
\(L(\frac12)\) \(\approx\) \(2.74763 - 2.60392i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (7.99e3 - 2.32e3i)T \)
good2 \( 1 + (-7.81 + 4.51i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (-10.9 + 18.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-44.4 - 25.6i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (121. - 210. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 232.T + 1.61e5T^{2} \)
13 \( 1 + (178. + 102. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-914. + 528. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.64e3 + 948. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 2.47e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.93e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.89e3iT - 2.86e7T^{2} \)
41 \( 1 + (-9.69e3 + 1.67e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 3.75e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.01e4T + 2.29e8T^{2} \)
53 \( 1 + (-862. - 1.49e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-4.08e3 + 2.35e3i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.87e3 + 2.81e3i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.36e4 - 2.36e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.30e4 - 3.99e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 2.06e4T + 2.07e9T^{2} \)
79 \( 1 + (-4.48e4 - 2.59e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.92e4 - 3.33e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (1.12e5 - 6.49e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 9.11e4iT - 8.58e9T^{2} \)
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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

−14.49309988041503064527056608669, −13.66435340697180221060895051578, −12.60475896705181954499803293734, −12.26981138267352461191858788629, −10.36962184057006484837867278250, −8.701714910196782890845246210406, −6.63183205736118263792196439423, −5.52157419205932466250900188694, −2.78913040993522511726749050568, −2.30238173175761111132803172388, 3.45751498741237670646191551058, 4.41112153595876840881044752954, 5.91997036317288050773580150370, 7.55492815064183569484365753246, 9.560063532904923768299774365716, 10.39449619755702168928770590608, 12.80952508192365644707212111785, 13.55139516213914640786912843558, 14.42168893104172925154297165840, 15.44666753122709224410525982265

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