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Properties

Label 2-91-91.89-c1-0-0
Degree $2$
Conductor $91$
Sign $0.631 - 0.775i$
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  + (1.14 + 1.14i)2-s + (−0.445 − 0.256i)3-s + 0.630i·4-s + (0.395 + 1.47i)5-s + (−0.215 − 0.805i)6-s + (0.0531 + 2.64i)7-s + (1.57 − 1.57i)8-s + (−1.36 − 2.36i)9-s + (−1.23 + 2.14i)10-s + (−0.745 − 2.78i)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 + 4.86·16-s − 6.21·17-s + ⋯
L(s)  = 1  + (0.811 + 0.811i)2-s + (−0.256 − 0.148i)3-s + 0.315i·4-s + (0.176 + 0.659i)5-s + (−0.0880 − 0.328i)6-s + (0.0200 + 0.999i)7-s + (0.555 − 0.555i)8-s + (−0.455 − 0.789i)9-s + (−0.391 + 0.678i)10-s + (−0.224 − 0.838i)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 + 1.21·16-s − 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19244 + 0.566948i\)
\(L(\frac12)\) \(\approx\) \(1.19244 + 0.566948i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0531 - 2.64i)T \)
13 \( 1 + (2.94 + 2.08i)T \)
good2 \( 1 + (-1.14 - 1.14i)T + 2iT^{2} \)
3 \( 1 + (0.445 + 0.256i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.395 - 1.47i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.745 + 2.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 6.21T + 17T^{2} \)
19 \( 1 + (-2.23 - 0.598i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.62iT - 23T^{2} \)
29 \( 1 + (-0.379 - 0.656i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.36 - 2.24i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
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 \( 1 + (2.13 - 0.571i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.47 - 4.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.623 + 0.623i)T + 59iT^{2} \)
61 \( 1 + (-4.48 + 2.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-15.1 + 4.06i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.3 - 2.76i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.80 + 6.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.24 - 7.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 - 1.51i)T - 83iT^{2} \)
89 \( 1 + (5.91 + 5.91i)T + 89iT^{2} \)
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

−14.34407042877020932907510531638, −13.45852359782526690882553165564, −12.29447615832936626994128818047, −11.24235362145702930600301202663, −9.903753684102686109723963168970, −8.500248738552353376758383394449, −6.91507802184371325167600685477, −6.06956312535044338881984514764, −5.09758784082592779211071326056, −3.07436699059344520828564349004, 2.32188013438093643962813516074, 4.41345720019145253183706956336, 4.94051656062411482412079196341, 7.01865361668705180321464290228, 8.382642133402848964771449804576, 9.983773738173181874553700361705, 10.93874809160506969076445866284, 11.88428144532680674839490791313, 12.95839088501259598250754621953, 13.61371062997082178605005499115

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