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Properties

Label 2-91-91.4-c1-0-3
Degree $2$
Conductor $91$
Sign $0.978 - 0.207i$
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  + 1.34i·2-s + (−1.02 − 1.77i)3-s + 0.190·4-s + (3.08 − 1.78i)5-s + (2.38 − 1.37i)6-s + (−2.44 + 1.00i)7-s + 2.94i·8-s + (−0.601 + 1.04i)9-s + (2.39 + 4.15i)10-s + (−1.10 + 0.639i)11-s + (−0.195 − 0.337i)12-s + (3.57 + 0.474i)13-s + (−1.35 − 3.29i)14-s + (−6.33 − 3.65i)15-s − 3.58·16-s − 7.73·17-s + ⋯
L(s)  = 1  + 0.951i·2-s + (−0.591 − 1.02i)3-s + 0.0951·4-s + (1.38 − 0.797i)5-s + (0.975 − 0.562i)6-s + (−0.924 + 0.381i)7-s + 1.04i·8-s + (−0.200 + 0.347i)9-s + (0.758 + 1.31i)10-s + (−0.333 + 0.192i)11-s + (−0.0563 − 0.0975i)12-s + (0.991 + 0.131i)13-s + (−0.362 − 0.879i)14-s + (−1.63 − 0.944i)15-s − 0.895·16-s − 1.87·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.978 - 0.207i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00827 + 0.105512i\)
\(L(\frac12)\) \(\approx\) \(1.00827 + 0.105512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.44 - 1.00i)T \)
13 \( 1 + (-3.57 - 0.474i)T \)
good2 \( 1 - 1.34iT - 2T^{2} \)
3 \( 1 + (1.02 + 1.77i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.08 + 1.78i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.10 - 0.639i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.73T + 17T^{2} \)
19 \( 1 + (-0.817 - 0.471i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 + (2.02 - 3.50i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 2.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.05iT - 37T^{2} \)
41 \( 1 + (3.63 + 2.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 - 3.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.774 + 0.447i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0399 + 0.0692i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + (-3.81 + 6.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.47 + 3.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.89 + 5.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.658 - 0.380i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.32iT - 83T^{2} \)
89 \( 1 - 7.57iT - 89T^{2} \)
97 \( 1 + (0.414 - 0.239i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.81241619090311161833155820063, −13.20248633330450587488642371985, −12.36804086327708484458608492270, −11.02714053215349499445358849360, −9.456472371352345130244845474237, −8.424009535353853819333600852407, −6.70313686942708721026070391838, −6.31295892817114879717802869343, −5.28908357404866682653632568992, −2.01020331761217424593089926431, 2.49236745136808444183187113798, 3.98474407140208452410106251378, 5.90320692866640025924775161771, 6.74073050418529605802363723009, 9.283899279345156254858956074495, 10.14402660686812818574045701141, 10.65846087884745674335167918857, 11.42879079674839048296064501369, 13.20116698633857230006912936007, 13.56342227544880503438459758459

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