Properties

Label 2-91-91.34-c1-0-1
Degree $2$
Conductor $91$
Sign $0.322 - 0.946i$
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  + (0.854 + 0.854i)2-s + 2.27i·3-s − 0.539i·4-s + (0.612 − 0.612i)5-s + (−1.94 + 1.94i)6-s + (−2.64 − 0.0148i)7-s + (2.17 − 2.17i)8-s − 2.17·9-s + 1.04·10-s + (−1.85 + 1.85i)11-s + 1.22·12-s + (0.104 − 3.60i)13-s + (−2.24 − 2.27i)14-s + (1.39 + 1.39i)15-s + 2.63·16-s + 3.04·17-s + ⋯
L(s)  = 1  + (0.604 + 0.604i)2-s + 1.31i·3-s − 0.269i·4-s + (0.274 − 0.274i)5-s + (−0.793 + 0.793i)6-s + (−0.999 − 0.00559i)7-s + (0.767 − 0.767i)8-s − 0.723·9-s + 0.331·10-s + (−0.559 + 0.559i)11-s + 0.353·12-s + (0.0289 − 0.999i)13-s + (−0.600 − 0.607i)14-s + (0.359 + 0.359i)15-s + 0.657·16-s + 0.739·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02873 + 0.736191i\)
\(L(\frac12)\) \(\approx\) \(1.02873 + 0.736191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.64 + 0.0148i)T \)
13 \( 1 + (-0.104 + 3.60i)T \)
good2 \( 1 + (-0.854 - 0.854i)T + 2iT^{2} \)
3 \( 1 - 2.27iT - 3T^{2} \)
5 \( 1 + (-0.612 + 0.612i)T - 5iT^{2} \)
11 \( 1 + (1.85 - 1.85i)T - 11iT^{2} \)
17 \( 1 - 3.04T + 17T^{2} \)
19 \( 1 + (0.104 - 0.104i)T - 19iT^{2} \)
23 \( 1 + 6.51iT - 23T^{2} \)
29 \( 1 + 3.78T + 29T^{2} \)
31 \( 1 + (6.77 - 6.77i)T - 31iT^{2} \)
37 \( 1 + (2.02 - 2.02i)T - 37iT^{2} \)
41 \( 1 + (-2.27 + 2.27i)T - 41iT^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + (5.21 + 5.21i)T + 47iT^{2} \)
53 \( 1 - 3.43T + 53T^{2} \)
59 \( 1 + (-9.15 - 9.15i)T + 59iT^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 + (1.04 + 1.04i)T + 67iT^{2} \)
71 \( 1 + (4.10 + 4.10i)T + 71iT^{2} \)
73 \( 1 + (6.92 + 6.92i)T + 73iT^{2} \)
79 \( 1 - 17.5T + 79T^{2} \)
83 \( 1 + (-10.5 + 10.5i)T - 83iT^{2} \)
89 \( 1 + (3.39 + 3.39i)T + 89iT^{2} \)
97 \( 1 + (4.44 - 4.44i)T - 97iT^{2} \)
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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

−14.75434818520951981564091630807, −13.31711264455927431218830362497, −12.59205165136006393270390865763, −10.51769696853620589098094123913, −10.15131351205321339863061734900, −9.079885749212406600343148139121, −7.23773285659281283319742971363, −5.74537599401323515740711861415, −4.92826436182125146014309806476, −3.51416850862159419630578358451, 2.18440271401638817364380283416, 3.61502473856282790590608499348, 5.75071352104688960147192043684, 7.00955796259508518278670336415, 7.985038331752409094351110607771, 9.586647008144001413717398296079, 11.10117982210607542121695971578, 12.04631984062965914554532225195, 12.94010816349095961769125670028, 13.47384125245627834339524690799

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