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Properties

Label 2-91-91.88-c1-0-4
Degree $2$
Conductor $91$
Sign $0.111 + 0.993i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.156 − 0.0904i)2-s − 1.82·3-s + (−0.983 − 1.70i)4-s + (2.32 − 1.34i)5-s + (0.285 + 0.165i)6-s + (−0.393 − 2.61i)7-s + 0.717i·8-s + 0.334·9-s − 0.485·10-s − 2.69i·11-s + (1.79 + 3.11i)12-s + (1.92 + 3.05i)13-s + (−0.174 + 0.445i)14-s + (−4.24 + 2.45i)15-s + (−1.90 + 3.29i)16-s + (2.38 + 4.12i)17-s + ⋯
L(s)  = 1  + (−0.110 − 0.0639i)2-s − 1.05·3-s + (−0.491 − 0.851i)4-s + (1.04 − 0.600i)5-s + (0.116 + 0.0673i)6-s + (−0.148 − 0.988i)7-s + 0.253i·8-s + 0.111·9-s − 0.153·10-s − 0.812i·11-s + (0.518 + 0.898i)12-s + (0.532 + 0.846i)13-s + (−0.0467 + 0.119i)14-s + (−1.09 + 0.633i)15-s + (−0.475 + 0.823i)16-s + (0.577 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513255 - 0.458933i\)
\(L(\frac12)\) \(\approx\) \(0.513255 - 0.458933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.393 + 2.61i)T \)
13 \( 1 + (-1.92 - 3.05i)T \)
good2 \( 1 + (0.156 + 0.0904i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + 1.82T + 3T^{2} \)
5 \( 1 + (-2.32 + 1.34i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.69iT - 11T^{2} \)
17 \( 1 + (-2.38 - 4.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.188iT - 19T^{2} \)
23 \( 1 + (-2.19 + 3.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.20 - 1.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.88 - 3.97i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.70 + 2.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.00 - 6.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.60 + 0.924i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.53 + 6.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.57 - 3.79i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.411T + 61T^{2} \)
67 \( 1 - 11.4iT - 67T^{2} \)
71 \( 1 + (-2.89 - 1.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (12.3 + 7.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.5iT - 83T^{2} \)
89 \( 1 + (5.10 + 2.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.390 - 0.225i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.70800309132219379623859191568, −13.02941109457624007359588862046, −11.43788682035109578819199127733, −10.57415300890009983978326577985, −9.709419675340659847087653857116, −8.522743174545387701838502027490, −6.34443582575627598222153188742, −5.76106261242398752756778389624, −4.44284558617138191126881327094, −1.12206179735262928995982259409, 2.89904003141645098837436661981, 5.13732214353981099604058925727, 6.05444989085280099395876855560, 7.42270441724624879988127557969, 9.003820631257257685057539513336, 9.946108813969285635971769541472, 11.24847750345295352554784943485, 12.25020383273604516758026303562, 13.06995735138602216876509740309, 14.22710399678867270359861867758

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