↑

Properties

Label 2-37-37.6-c6-0-17
Degree $2$
Conductor $37$
Sign $-0.933 + 0.359i$
Analytic cond. $8.51200$
Root an. cond. $2.91753$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.39 − 9.39i)2-s − 33.9i·3-s − 112. i·4-s + (19.7 + 19.7i)5-s + (−318. − 318. i)6-s + 150.·7-s + (−454. − 454. i)8-s − 421.·9-s + 371.·10-s + 1.14e3i·11-s − 3.81e3·12-s + (−678. − 678. i)13-s + (1.41e3 − 1.41e3i)14-s + (671. − 671. i)15-s − 1.34e3·16-s + (4.20e3 + 4.20e3i)17-s + ⋯
L(s)  = 1  + (1.17 − 1.17i)2-s − 1.25i·3-s − 1.75i·4-s + (0.158 + 0.158i)5-s + (−1.47 − 1.47i)6-s + 0.439·7-s + (−0.887 − 0.887i)8-s − 0.578·9-s + 0.371·10-s + 0.856i·11-s − 2.20·12-s + (−0.308 − 0.308i)13-s + (0.516 − 0.516i)14-s + (0.198 − 0.198i)15-s − 0.328·16-s + (0.855 + 0.855i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(8.51200\)
Root analytic conductor: \(2.91753\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.581348 - 3.12542i\)
\(L(\frac12)\) \(\approx\) \(0.581348 - 3.12542i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-4.78e4 - 1.66e4i)T \)
good2 \( 1 + (-9.39 + 9.39i)T - 64iT^{2} \)
3 \( 1 + 33.9iT - 729T^{2} \)
5 \( 1 + (-19.7 - 19.7i)T + 1.56e4iT^{2} \)
7 \( 1 - 150.T + 1.17e5T^{2} \)
11 \( 1 - 1.14e3iT - 1.77e6T^{2} \)
13 \( 1 + (678. + 678. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-4.20e3 - 4.20e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (5.67e3 + 5.67e3i)T + 4.70e7iT^{2} \)
23 \( 1 + (-2.02e3 - 2.02e3i)T + 1.48e8iT^{2} \)
29 \( 1 + (3.54e3 - 3.54e3i)T - 5.94e8iT^{2} \)
31 \( 1 + (1.50e3 - 1.50e3i)T - 8.87e8iT^{2} \)
41 \( 1 - 1.28e5iT - 4.75e9T^{2} \)
43 \( 1 + (-3.79e4 - 3.79e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 1.21e5T + 1.07e10T^{2} \)
53 \( 1 + 1.01e5T + 2.21e10T^{2} \)
59 \( 1 + (2.45e5 + 2.45e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.68e5 - 1.68e5i)T - 5.15e10iT^{2} \)
67 \( 1 - 4.22e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.49e4T + 1.28e11T^{2} \)
73 \( 1 - 4.11e5iT - 1.51e11T^{2} \)
79 \( 1 + (-1.86e5 - 1.86e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 1.99e5T + 3.26e11T^{2} \)
89 \( 1 + (-2.35e5 + 2.35e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (5.09e5 + 5.09e5i)T + 8.32e11iT^{2} \)
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

−14.18494169271382921310991985303, −12.95154360601900454281828216904, −12.51513983155639787469161328894, −11.34848563276440094188387010029, −10.07004700279498310019780393161, −7.83290532848786483831263799995, −6.23885230041046761921661567448, −4.56711995995340185187945428406, −2.55118821739186792327170734866, −1.33880773822604507469631043260, 3.59203466557613246714771507543, 4.80109910906461203105371963032, 5.84079340127379441084204468179, 7.61853225039437184455261105686, 9.174823112176054756587686438667, 10.74987903171538856448583568121, 12.31211940876310211357370691365, 13.77887982487658653995154147351, 14.60178603926266677051714871327, 15.49741157678071781223066644888

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