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Properties

Label 2-91-13.4-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.0340 - 0.999i$
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  + (0.713 + 0.411i)2-s + (−1.33 + 2.30i)3-s + (−0.660 − 1.14i)4-s + 3.16i·5-s + (−1.89 + 1.09i)6-s + (0.866 − 0.5i)7-s − 2.73i·8-s + (−2.03 − 3.53i)9-s + (−1.30 + 2.25i)10-s + (5.14 + 2.97i)11-s + 3.51·12-s + (−0.0766 − 3.60i)13-s + 0.823·14-s + (−7.28 − 4.20i)15-s + (−0.195 + 0.338i)16-s + (−1.34 − 2.33i)17-s + ⋯
L(s)  = 1  + (0.504 + 0.291i)2-s + (−0.767 + 1.33i)3-s + (−0.330 − 0.572i)4-s + 1.41i·5-s + (−0.774 + 0.447i)6-s + (0.327 − 0.188i)7-s − 0.967i·8-s + (−0.679 − 1.17i)9-s + (−0.411 + 0.713i)10-s + (1.55 + 0.895i)11-s + 1.01·12-s + (−0.0212 − 0.999i)13-s + 0.220·14-s + (−1.88 − 1.08i)15-s + (−0.0488 + 0.0845i)16-s + (−0.327 − 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0340 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0340 - 0.999i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0340 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681217 + 0.704835i\)
\(L(\frac12)\) \(\approx\) \(0.681217 + 0.704835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (0.0766 + 3.60i)T \)
good2 \( 1 + (-0.713 - 0.411i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.33 - 2.30i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.16iT - 5T^{2} \)
11 \( 1 + (-5.14 - 2.97i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.34 + 2.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 0.978i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.36 - 2.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 + (5.63 + 3.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.23 + 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.49 - 6.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.456iT - 47T^{2} \)
53 \( 1 - 0.399T + 53T^{2} \)
59 \( 1 + (-4.16 + 2.40i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.578 - 1.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 + 3.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.90 + 2.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.30iT - 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 + 6.19iT - 83T^{2} \)
89 \( 1 + (-3.08 - 1.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.96 - 1.71i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.66625402390913518144048519476, −13.77438978196193640930151428532, −11.95453800125988746864020045301, −10.95034129633082295264378016602, −10.13575811931214917327827468830, −9.421178181167146303666712323804, −7.08122048986503234780724096658, −6.03414898312384266644644028520, −4.78596105179415781043020818011, −3.69282561723906964847530201773, 1.47858196731426674660265851344, 4.16000571080917917978018388091, 5.48067124620818944968184865505, 6.76658255659835676260926153882, 8.354561653963263044528692884816, 8.946566539491706663507915394209, 11.38460470651188933935368924562, 12.02097108709856743664239871143, 12.53554047400001471824173567196, 13.55616700450684359645343244630

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