↑

Properties

Label 2-37-37.31-c8-0-11
Degree $2$
Conductor $37$
Sign $-0.993 + 0.111i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (22.1 + 22.1i)2-s + 81.8i·3-s + 728. i·4-s + (618. − 618. i)5-s + (−1.81e3 + 1.81e3i)6-s − 2.92e3·7-s + (−1.04e4 + 1.04e4i)8-s − 144.·9-s + 2.74e4·10-s + 1.74e4i·11-s − 5.96e4·12-s + (2.42e4 − 2.42e4i)13-s + (−6.49e4 − 6.49e4i)14-s + (5.06e4 + 5.06e4i)15-s − 2.79e5·16-s + (3.88e4 − 3.88e4i)17-s + ⋯
L(s)  = 1  + (1.38 + 1.38i)2-s + 1.01i·3-s + 2.84i·4-s + (0.990 − 0.990i)5-s + (−1.40 + 1.40i)6-s − 1.21·7-s + (−2.56 + 2.56i)8-s − 0.0220·9-s + 2.74·10-s + 1.19i·11-s − 2.87·12-s + (0.849 − 0.849i)13-s + (−1.69 − 1.69i)14-s + (1.00 + 1.00i)15-s − 4.25·16-s + (0.464 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.993 + 0.111i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ -0.993 + 0.111i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.216843 - 3.88006i\)
\(L(\frac12)\) \(\approx\) \(0.216843 - 3.88006i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.28e6 - 1.36e6i)T \)
good2 \( 1 + (-22.1 - 22.1i)T + 256iT^{2} \)
3 \( 1 - 81.8iT - 6.56e3T^{2} \)
5 \( 1 + (-618. + 618. i)T - 3.90e5iT^{2} \)
7 \( 1 + 2.92e3T + 5.76e6T^{2} \)
11 \( 1 - 1.74e4iT - 2.14e8T^{2} \)
13 \( 1 + (-2.42e4 + 2.42e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-3.88e4 + 3.88e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (-4.56e4 + 4.56e4i)T - 1.69e10iT^{2} \)
23 \( 1 + (-1.57e4 + 1.57e4i)T - 7.83e10iT^{2} \)
29 \( 1 + (-4.78e5 - 4.78e5i)T + 5.00e11iT^{2} \)
31 \( 1 + (-5.64e5 - 5.64e5i)T + 8.52e11iT^{2} \)
41 \( 1 + 2.79e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.12e6 + 2.12e6i)T - 1.16e13iT^{2} \)
47 \( 1 - 1.33e6T + 2.38e13T^{2} \)
53 \( 1 + 9.44e6T + 6.22e13T^{2} \)
59 \( 1 + (-1.11e7 + 1.11e7i)T - 1.46e14iT^{2} \)
61 \( 1 + (-5.64e6 - 5.64e6i)T + 1.91e14iT^{2} \)
67 \( 1 + 1.47e5iT - 4.06e14T^{2} \)
71 \( 1 + 5.52e6T + 6.45e14T^{2} \)
73 \( 1 - 2.58e7iT - 8.06e14T^{2} \)
79 \( 1 + (-3.64e6 + 3.64e6i)T - 1.51e15iT^{2} \)
83 \( 1 - 2.98e7T + 2.25e15T^{2} \)
89 \( 1 + (4.83e7 + 4.83e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (-1.14e8 + 1.14e8i)T - 7.83e15iT^{2} \)
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

−15.60961212790107100004604979107, −14.08717958008937363484617472018, −13.03109178148540408205860224089, −12.45199458301575347578259299367, −9.931725600364369390658238546352, −8.800822727066713956205724215352, −6.93635258907684919868534192798, −5.58368372661292989307184554339, −4.70241943290178561918101003314, −3.28399797208218820622964547272, 1.08881875676091169556096425877, 2.45051180742094814870898866182, 3.57945367858066938128555053373, 6.11694677640550948537859682888, 6.37184017830970849069163580115, 9.604969848842316014736330569809, 10.59868279070163544743546685247, 11.79794574418167430594236402714, 13.02586770907885572792776604453, 13.62769440483242531870548267981

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