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Properties

Label 2-91-91.19-c1-0-2
Degree $2$
Conductor $91$
Sign $0.837 + 0.545i$
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  + (−1.56 − 0.419i)2-s + 0.513i·3-s + (0.546 + 0.315i)4-s + (1.47 − 0.395i)5-s + (0.215 − 0.805i)6-s + (1.27 − 2.31i)7-s + (1.57 + 1.57i)8-s + 2.73·9-s − 2.47·10-s + (−2.03 − 2.03i)11-s + (−0.162 + 0.280i)12-s + (2.94 − 2.08i)13-s + (−2.97 + 3.09i)14-s + (0.203 + 0.757i)15-s + (−2.43 − 4.21i)16-s + (−3.10 + 5.38i)17-s + ⋯
L(s)  = 1  + (−1.10 − 0.296i)2-s + 0.296i·3-s + (0.273 + 0.157i)4-s + (0.659 − 0.176i)5-s + (0.0880 − 0.328i)6-s + (0.482 − 0.875i)7-s + (0.555 + 0.555i)8-s + 0.911·9-s − 0.783·10-s + (−0.614 − 0.614i)11-s + (−0.0467 + 0.0810i)12-s + (0.816 − 0.577i)13-s + (−0.794 + 0.827i)14-s + (0.0524 + 0.195i)15-s + (−0.607 − 1.05i)16-s + (−0.753 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.837 + 0.545i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.837 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629758 - 0.187018i\)
\(L(\frac12)\) \(\approx\) \(0.629758 - 0.187018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.27 + 2.31i)T \)
13 \( 1 + (-2.94 + 2.08i)T \)
good2 \( 1 + (1.56 + 0.419i)T + (1.73 + i)T^{2} \)
3 \( 1 - 0.513iT - 3T^{2} \)
5 \( 1 + (-1.47 + 0.395i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.03 + 2.03i)T + 11iT^{2} \)
17 \( 1 + (3.10 - 5.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 1.63i)T + 19iT^{2} \)
23 \( 1 + (4.86 - 2.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.379 + 0.656i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.24 + 8.36i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.56 - 5.82i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.94 + 0.522i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.24 - 1.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.571 + 2.13i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.47 - 4.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.228 - 0.851i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 5.17iT - 61T^{2} \)
67 \( 1 + (11.1 - 11.1i)T - 67iT^{2} \)
71 \( 1 + (10.3 + 2.76i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.72 - 1.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.24 - 7.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.51 - 1.51i)T + 83iT^{2} \)
89 \( 1 + (8.08 + 2.16i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.933 + 3.48i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.66911576080684186399377126027, −13.25849461593332528032400284269, −11.33363589939775844021715564022, −10.37519924813528779397001486354, −9.909769474454919410529520902986, −8.508989954761619216774891754863, −7.66002377409050093017565560616, −5.80344458447429573485324720865, −4.16840067794771900282242357032, −1.53053625006290779582224790493, 1.97105197489096490057304707290, 4.71079133922441173577319243247, 6.47988469027670881603100599059, 7.53537197876985461209202556005, 8.744575055620273928503921917587, 9.608229075990084718517360948524, 10.60458522776038280977122265110, 12.02696652539369362440880940299, 13.21031976382428187667547263439, 14.08928965066931106915399758000

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