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Properties

Label 2-91-91.59-c1-0-2
Degree $2$
Conductor $91$
Sign $0.411 - 0.911i$
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  + (1.51 + 1.51i)2-s + (0.0170 − 0.00986i)3-s + 2.58i·4-s + (−1.34 − 0.360i)5-s + (0.0408 + 0.0109i)6-s + (−0.246 − 2.63i)7-s + (−0.893 + 0.893i)8-s + (−1.49 + 2.59i)9-s + (−1.49 − 2.58i)10-s + (0.336 + 0.0902i)11-s + (0.0255 + 0.0442i)12-s + (−1.32 − 3.35i)13-s + (3.61 − 4.36i)14-s + (−0.0265 + 0.00711i)15-s + 2.47·16-s − 0.982·17-s + ⋯
L(s)  = 1  + (1.07 + 1.07i)2-s + (0.00986 − 0.00569i)3-s + 1.29i·4-s + (−0.601 − 0.161i)5-s + (0.0166 + 0.00446i)6-s + (−0.0931 − 0.995i)7-s + (−0.315 + 0.315i)8-s + (−0.499 + 0.865i)9-s + (−0.471 − 0.816i)10-s + (0.101 + 0.0272i)11-s + (0.00737 + 0.0127i)12-s + (−0.368 − 0.929i)13-s + (0.966 − 1.16i)14-s + (−0.00685 + 0.00183i)15-s + 0.618·16-s − 0.238·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22369 + 0.790211i\)
\(L(\frac12)\) \(\approx\) \(1.22369 + 0.790211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.246 + 2.63i)T \)
13 \( 1 + (1.32 + 3.35i)T \)
good2 \( 1 + (-1.51 - 1.51i)T + 2iT^{2} \)
3 \( 1 + (-0.0170 + 0.00986i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.34 + 0.360i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.336 - 0.0902i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 0.982T + 17T^{2} \)
19 \( 1 + (-1.19 - 4.45i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.30iT - 23T^{2} \)
29 \( 1 + (0.941 - 1.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.755 + 2.81i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 + (-0.580 - 2.16i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.47 - 3.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.83 - 10.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.77 + 6.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.7 - 10.7i)T + 59iT^{2} \)
61 \( 1 + (5.59 + 3.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.61 + 9.74i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.43 + 9.07i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.3 + 2.76i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.890 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.33 + 8.33i)T - 83iT^{2} \)
89 \( 1 + (9.61 + 9.61i)T + 89iT^{2} \)
97 \( 1 + (12.6 + 3.39i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.29965675175053065600390146460, −13.47157118695466380275193218294, −12.62237643159149152599099101467, −11.26778625223606725126656472014, −10.02834446717114380307817247202, −7.961491412903571384519283918557, −7.54446412090557290603579707625, −6.01209720151401206741922800401, −4.81627956964106617792819775825, −3.60576789321904206018175183160, 2.53079074764762395171028590956, 3.84169718357032494400233042602, 5.21704747110490677363685528649, 6.66807396200833824697450588181, 8.558385001763932231875887414636, 9.729314021836038823663308547369, 11.37984130138378655596993796520, 11.71618283952044386495023499098, 12.59967912213413776508211126976, 13.74793766421791840004837298582

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