Properties

Label 2-37-37.7-c7-0-0
Degree $2$
Conductor $37$
Sign $-0.209 - 0.977i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (14.2 − 5.17i)2-s + (−80.6 − 29.3i)3-s + (77.2 − 64.7i)4-s + (−48.8 − 277. i)5-s − 1.29e3·6-s + (203. + 1.15e3i)7-s + (−205. + 356. i)8-s + (3.96e3 + 3.32e3i)9-s + (−2.12e3 − 3.68e3i)10-s + (499. − 864. i)11-s + (−8.12e3 + 2.95e3i)12-s + (−5.87e3 + 4.92e3i)13-s + (8.86e3 + 1.53e4i)14-s + (−4.19e3 + 2.37e4i)15-s + (−3.32e3 + 1.88e4i)16-s + (−2.44e4 − 2.05e4i)17-s + ⋯
L(s)  = 1  + (1.25 − 0.457i)2-s + (−1.72 − 0.627i)3-s + (0.603 − 0.506i)4-s + (−0.174 − 0.991i)5-s − 2.45·6-s + (0.224 + 1.27i)7-s + (−0.142 + 0.246i)8-s + (1.81 + 1.52i)9-s + (−0.673 − 1.16i)10-s + (0.113 − 0.195i)11-s + (−1.35 + 0.494i)12-s + (−0.741 + 0.622i)13-s + (0.863 + 1.49i)14-s + (−0.320 + 1.81i)15-s + (−0.202 + 1.14i)16-s + (−1.20 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.208566 + 0.257851i\)
\(L(\frac12)\) \(\approx\) \(0.208566 + 0.257851i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.78e5 + 2.51e5i)T \)
good2 \( 1 + (-14.2 + 5.17i)T + (98.0 - 82.2i)T^{2} \)
3 \( 1 + (80.6 + 29.3i)T + (1.67e3 + 1.40e3i)T^{2} \)
5 \( 1 + (48.8 + 277. i)T + (-7.34e4 + 2.67e4i)T^{2} \)
7 \( 1 + (-203. - 1.15e3i)T + (-7.73e5 + 2.81e5i)T^{2} \)
11 \( 1 + (-499. + 864. i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (5.87e3 - 4.92e3i)T + (1.08e7 - 6.17e7i)T^{2} \)
17 \( 1 + (2.44e4 + 2.05e4i)T + (7.12e7 + 4.04e8i)T^{2} \)
19 \( 1 + (2.94e4 + 1.07e4i)T + (6.84e8 + 5.74e8i)T^{2} \)
23 \( 1 + (-3.07e4 - 5.33e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (6.69e4 - 1.15e5i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 - 1.03e4T + 2.75e10T^{2} \)
41 \( 1 + (5.08e5 - 4.26e5i)T + (3.38e10 - 1.91e11i)T^{2} \)
43 \( 1 + 5.21e5T + 2.71e11T^{2} \)
47 \( 1 + (-5.72e4 - 9.91e4i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-3.01e5 + 1.71e6i)T + (-1.10e12 - 4.01e11i)T^{2} \)
59 \( 1 + (7.95e4 - 4.50e5i)T + (-2.33e12 - 8.51e11i)T^{2} \)
61 \( 1 + (8.20e5 - 6.88e5i)T + (5.45e11 - 3.09e12i)T^{2} \)
67 \( 1 + (4.19e5 + 2.37e6i)T + (-5.69e12 + 2.07e12i)T^{2} \)
71 \( 1 + (-2.46e6 - 8.97e5i)T + (6.96e12 + 5.84e12i)T^{2} \)
73 \( 1 + 2.31e6T + 1.10e13T^{2} \)
79 \( 1 + (-1.12e5 - 6.40e5i)T + (-1.80e13 + 6.56e12i)T^{2} \)
83 \( 1 + (-7.58e6 - 6.36e6i)T + (4.71e12 + 2.67e13i)T^{2} \)
89 \( 1 + (-1.51e6 + 8.58e6i)T + (-4.15e13 - 1.51e13i)T^{2} \)
97 \( 1 + (1.69e6 + 2.92e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.16432495374321831526387953825, −13.36268384497068948005270458869, −12.57563901931339165371870591227, −11.82771319999694880875173021853, −11.21755749674515884949552843549, −8.884847514842057867209988231295, −6.66457325678817363312903923831, −5.26036174839839569968956070959, −4.78919206061087682052236503809, −1.95125907009890781779452418084, 0.11641421392064795708021181918, 3.90851408440804015666191104384, 4.80298929806854621214390075772, 6.31117274174633954453985732653, 7.01862279267617305514650473995, 10.29010963721508696606093876805, 10.81110731893729044687530875777, 12.15581897913508531449860667209, 13.28727916517530226672666785474, 14.85681378924974890265652013284

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