↑

Properties

Label 2-91-13.12-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.410 - 0.911i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
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  + 2.48i·2-s + 1.67·3-s − 4.15·4-s − 0.675i·5-s + 4.15i·6-s + i·7-s − 5.35i·8-s − 0.193·9-s + 1.67·10-s − 4.48i·11-s − 6.96·12-s + (3.28 − 1.48i)13-s − 2.48·14-s − 1.13i·15-s + 4.96·16-s − 3.28·17-s + ⋯
L(s)  = 1  + 1.75i·2-s + 0.967·3-s − 2.07·4-s − 0.301i·5-s + 1.69i·6-s + 0.377i·7-s − 1.89i·8-s − 0.0646·9-s + 0.529·10-s − 1.35i·11-s − 2.00·12-s + (0.911 − 0.410i)13-s − 0.663·14-s − 0.292i·15-s + 1.24·16-s − 0.797·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.410 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615013 + 0.951679i\)
\(L(\frac12)\) \(\approx\) \(0.615013 + 0.951679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
13 \( 1 + (-3.28 + 1.48i)T \)
good2 \( 1 - 2.48iT - 2T^{2} \)
3 \( 1 - 1.67T + 3T^{2} \)
5 \( 1 + 0.675iT - 5T^{2} \)
11 \( 1 + 4.48iT - 11T^{2} \)
17 \( 1 + 3.28T + 17T^{2} \)
19 \( 1 - 5.21iT - 19T^{2} \)
23 \( 1 - 4.76T + 23T^{2} \)
29 \( 1 + 9.31T + 29T^{2} \)
31 \( 1 + 1.63iT - 31T^{2} \)
37 \( 1 - 1.44iT - 37T^{2} \)
41 \( 1 - 7.92iT - 41T^{2} \)
43 \( 1 + 4.61T + 43T^{2} \)
47 \( 1 + 7.86iT - 47T^{2} \)
53 \( 1 + 3.15T + 53T^{2} \)
59 \( 1 - 2.54iT - 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 7.35iT - 67T^{2} \)
71 \( 1 + 7.75iT - 71T^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 1.45iT - 83T^{2} \)
89 \( 1 - 7.79iT - 89T^{2} \)
97 \( 1 - 17.9iT - 97T^{2} \)
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

−14.70246133465779420293577137426, −13.64424879761094911846614659756, −13.10268496781018272864341576876, −11.14806387964268816168984566564, −9.233992423511432531015367765288, −8.571885557009975947464026812881, −7.926224027525450607697817132418, −6.35649436484857851357481458100, −5.37440888506246759159074062061, −3.53525046288720474932622155674, 2.06041210560032184859029994599, 3.34558627209866481755641184788, 4.60458169138919007256741610810, 7.13783554861322453754111723515, 8.841679912133318356206106547534, 9.383600005778452324614398126001, 10.73038186074594769689269652180, 11.38545309976043029653588784944, 12.81393082670946613469479312458, 13.41864048270695316227915456715

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