↑

Properties

Label 2-371-371.23-c0-0-1
Degree $2$
Conductor $371$
Sign $0.948 + 0.315i$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)4-s + (1.36 − 0.366i)5-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)9-s + (−0.866 + 0.5i)11-s − 13-s + (0.499 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.366 + 1.36i)19-s + (0.999 − i)20-s + (0.866 − 0.5i)25-s + (−0.499 + 0.866i)28-s − i·29-s + (1.36 + 0.366i)31-s + (−0.999 + i)35-s − 0.999·36-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (1.36 − 0.366i)5-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)9-s + (−0.866 + 0.5i)11-s − 13-s + (0.499 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.366 + 1.36i)19-s + (0.999 − i)20-s + (0.866 − 0.5i)25-s + (−0.499 + 0.866i)28-s − i·29-s + (1.36 + 0.366i)31-s + (−0.999 + i)35-s − 0.999·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $0.948 + 0.315i$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ 0.948 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9793219402\)
\(L(\frac12)\) \(\approx\) \(0.9793219402\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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

−11.70185642154793826560423350740, −10.27359576751646233224707313451, −9.946317799401421627229614432593, −9.081404010350004521780305764699, −7.79056944680626222685581877263, −6.34787127005326191660155454435, −6.02711853237632161353968381548, −5.03089747246438883281880752767, −2.88400724639667910878189605563, −2.05325363447704727572393283031, 2.55927040388480288522659849239, 2.84509302734454608021851253508, 4.96414616267858455465499321229, 6.08537004886212876502876075071, 6.81809767740737543445254826511, 7.74859096958008180642820370318, 9.031301586884312440229129736237, 9.954404847797729708524356859867, 10.78149640391842362311890290087, 11.42820317685250272515291342800

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