↑

Properties

Label 2-370-185.82-c1-0-15
Degree $2$
Conductor $370$
Sign $-0.618 + 0.785i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.465 − 0.124i)3-s + (−0.499 + 0.866i)4-s + (−2.23 + 0.126i)5-s + (−0.341 − 0.341i)6-s + (1.90 − 0.510i)7-s + 0.999·8-s + (−2.39 + 1.38i)9-s + (1.22 + 1.87i)10-s − 5.36i·11-s + (−0.124 + 0.465i)12-s + (3.28 − 5.68i)13-s + (−1.39 − 1.39i)14-s + (−1.02 + 0.337i)15-s + (−0.5 − 0.866i)16-s + (−3.27 + 1.89i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.269 − 0.0720i)3-s + (−0.249 + 0.433i)4-s + (−0.998 + 0.0566i)5-s + (−0.139 − 0.139i)6-s + (0.720 − 0.192i)7-s + 0.353·8-s + (−0.798 + 0.461i)9-s + (0.387 + 0.591i)10-s − 1.61i·11-s + (−0.0360 + 0.134i)12-s + (0.910 − 1.57i)13-s + (−0.372 − 0.372i)14-s + (−0.264 + 0.0871i)15-s + (−0.125 − 0.216i)16-s + (−0.794 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.618 + 0.785i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364880 - 0.751359i\)
\(L(\frac12)\) \(\approx\) \(0.364880 - 0.751359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (2.23 - 0.126i)T \)
37 \( 1 + (6.05 - 0.586i)T \)
good3 \( 1 + (-0.465 + 0.124i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.90 + 0.510i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 + (-3.28 + 5.68i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.27 - 1.89i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.79 + 6.69i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.59T + 23T^{2} \)
29 \( 1 + (-3.86 - 3.86i)T + 29iT^{2} \)
31 \( 1 + (-4.40 + 4.40i)T - 31iT^{2} \)
41 \( 1 + (-3.09 - 1.78i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (-2.56 - 2.56i)T + 47iT^{2} \)
53 \( 1 + (6.08 + 1.62i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.19 - 1.12i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.33 + 12.4i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.30 - 4.86i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.00 + 5.20i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.42 - 6.42i)T + 73iT^{2} \)
79 \( 1 + (-1.70 - 6.37i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-5.79 - 1.55i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-4.07 + 15.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + 0.635iT - 97T^{2} \)
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

−10.93639367689819548316875326512, −10.73278830147646706723063438710, −8.873991543846699524547826499271, −8.278260436808316152123536109027, −7.907246206242927750027273938415, −6.27465200641855789207905666414, −4.95348287285323083401761173660, −3.64072341465846628689937124646, −2.73122564591394711019025850266, −0.62563951065493071927906137742, 1.92839108571936373477610073617, 3.96483252985830155887944247844, 4.66829714349340975451621231656, 6.20055685659284316688579227067, 7.09621923634898548763076516378, 8.155925858801097306431577318352, 8.689278934487666104910010081049, 9.644524157825396110450671033383, 10.81310895847640131058080832549, 11.87285789680222851147336731673

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