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Properties

Label 2-91-91.54-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.841 - 0.539i$
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  + (−0.347 + 0.347i)2-s + (−2.11 − 1.22i)3-s + 1.75i·4-s + (−3.47 + 0.931i)5-s + (1.15 − 0.310i)6-s + (0.701 + 2.55i)7-s + (−1.30 − 1.30i)8-s + (1.49 + 2.58i)9-s + (0.883 − 1.52i)10-s + (−2.04 + 0.547i)11-s + (2.15 − 3.72i)12-s + (0.582 − 3.55i)13-s + (−1.12 − 0.642i)14-s + (8.50 + 2.27i)15-s − 2.61·16-s + 1.14·17-s + ⋯
L(s)  = 1  + (−0.245 + 0.245i)2-s + (−1.22 − 0.706i)3-s + 0.879i·4-s + (−1.55 + 0.416i)5-s + (0.473 − 0.126i)6-s + (0.264 + 0.964i)7-s + (−0.461 − 0.461i)8-s + (0.496 + 0.860i)9-s + (0.279 − 0.483i)10-s + (−0.616 + 0.165i)11-s + (0.620 − 1.07i)12-s + (0.161 − 0.986i)13-s + (−0.301 − 0.171i)14-s + (2.19 + 0.588i)15-s − 0.653·16-s + 0.277·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.841 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0722141 + 0.246367i\)
\(L(\frac12)\) \(\approx\) \(0.0722141 + 0.246367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.701 - 2.55i)T \)
13 \( 1 + (-0.582 + 3.55i)T \)
good2 \( 1 + (0.347 - 0.347i)T - 2iT^{2} \)
3 \( 1 + (2.11 + 1.22i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.47 - 0.931i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.04 - 0.547i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 + (1.18 - 4.40i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 1.48iT - 23T^{2} \)
29 \( 1 + (-2.75 - 4.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.56 - 5.85i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.91 + 1.91i)T + 37iT^{2} \)
41 \( 1 + (1.21 - 4.54i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.55 + 2.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.74 + 6.51i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.74 + 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.08 - 8.08i)T - 59iT^{2} \)
61 \( 1 + (-8.20 + 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.99 - 7.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.74 - 6.52i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.46 - 1.46i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.91 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.06 - 5.06i)T + 83iT^{2} \)
89 \( 1 + (-7.05 + 7.05i)T - 89iT^{2} \)
97 \( 1 + (13.1 - 3.52i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.87784806656388990477350602067, −12.84854729068290890660270362266, −12.23587113465286260045974990196, −11.67194902280341276075863700406, −10.62480417578492293588837700167, −8.449335959290084047162348299970, −7.74987179683288897285721796488, −6.73021792293595183503526634419, −5.26502815528373643790148836392, −3.34847397081663188355285059072, 0.38484476567988772685278203034, 4.22672697165845383695426464427, 4.98514190440739179428077787225, 6.58899748139106316725201606697, 8.071009567513003310032238248543, 9.578773293060048076612153411545, 10.82973368286009416089938936495, 11.18538800571783471044024845382, 12.06345329984051904868291927267, 13.68646678183613688944449254444

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