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Properties

Label 2-91-91.30-c1-0-0
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: $\chi_{91} (30, \cdot )$
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

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

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