Properties

Label 2-37-37.10-c5-0-11
Degree $2$
Conductor $37$
Sign $-0.443 + 0.896i$
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  + (0.684 − 1.18i)2-s + (−2.47 − 4.28i)3-s + (15.0 + 26.0i)4-s + (−49.2 − 85.3i)5-s − 6.77·6-s + (−21.8 − 37.8i)7-s + 85.0·8-s + (109. − 189. i)9-s − 134.·10-s − 339.·11-s + (74.4 − 129. i)12-s + (−523. − 906. i)13-s − 59.8·14-s + (−243. + 421. i)15-s + (−423. + 734. i)16-s + (594. − 1.03e3i)17-s + ⋯
L(s)  = 1  + (0.121 − 0.209i)2-s + (−0.158 − 0.274i)3-s + (0.470 + 0.815i)4-s + (−0.881 − 1.52i)5-s − 0.0767·6-s + (−0.168 − 0.291i)7-s + 0.469·8-s + (0.449 − 0.778i)9-s − 0.426·10-s − 0.847·11-s + (0.149 − 0.258i)12-s + (−0.859 − 1.48i)13-s − 0.0815·14-s + (−0.279 + 0.484i)15-s + (−0.413 + 0.716i)16-s + (0.499 − 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.443 + 0.896i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.443 + 0.896i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.659568 - 1.06179i\)
\(L(\frac12)\) \(\approx\) \(0.659568 - 1.06179i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-8.29e3 - 687. i)T \)
good2 \( 1 + (-0.684 + 1.18i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (2.47 + 4.28i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (49.2 + 85.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (21.8 + 37.8i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 339.T + 1.61e5T^{2} \)
13 \( 1 + (523. + 906. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-594. + 1.03e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.35e3 - 2.35e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 1.40e3T + 6.43e6T^{2} \)
29 \( 1 - 4.86e3T + 2.05e7T^{2} \)
31 \( 1 - 564.T + 2.86e7T^{2} \)
41 \( 1 + (4.74e3 + 8.21e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.85e4T + 1.47e8T^{2} \)
47 \( 1 - 3.09e3T + 2.29e8T^{2} \)
53 \( 1 + (-2.67e3 + 4.63e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-9.41e3 + 1.63e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.33e3 - 1.27e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.72e4 + 4.71e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-2.50e4 - 4.33e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 3.95e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.94e4 - 5.09e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.30e4 + 7.45e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-8.33e3 + 1.44e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 2.79e4T + 8.58e9T^{2} \)
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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

−15.36582340322895803594878335694, −13.20543814173187135412389481803, −12.41023906714572254201884847116, −11.88965175710606744993376352138, −9.995234996212755634845103515186, −8.187872101265147896608421611982, −7.43780809435101414227235824628, −5.09871882421870244835979043170, −3.43399630281417040024984412758, −0.71392663875005018034831597665, 2.58416547105116065154773440218, 4.79518492330805917867074503051, 6.64475487087819039264828104091, 7.53951554486386600664307583148, 9.900041574249049021431955886033, 10.84827226652190737589763145677, 11.69692873584372371084695282636, 13.72338845112807330330682479566, 14.86825624755367587814553206011, 15.51945097632411637175832612647

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