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Properties

Label 2-91-7.4-c1-0-1
Degree $2$
Conductor $91$
Sign $0.386 - 0.922i$
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  + (0.190 + 0.330i)2-s + (−1.11 + 1.93i)3-s + (0.927 − 1.60i)4-s + (1.11 + 1.93i)5-s − 0.854·6-s + (−2 + 1.73i)7-s + 1.47·8-s + (−1 − 1.73i)9-s + (−0.427 + 0.739i)10-s + (1.5 − 2.59i)11-s + (2.07 + 3.59i)12-s − 13-s + (−0.954 − 0.330i)14-s − 5.00·15-s + (−1.57 − 2.72i)16-s + (3.73 − 6.47i)17-s + ⋯
L(s)  = 1  + (0.135 + 0.233i)2-s + (−0.645 + 1.11i)3-s + (0.463 − 0.802i)4-s + (0.499 + 0.866i)5-s − 0.348·6-s + (−0.755 + 0.654i)7-s + 0.520·8-s + (−0.333 − 0.577i)9-s + (−0.135 + 0.233i)10-s + (0.452 − 0.783i)11-s + (0.598 + 1.03i)12-s − 0.277·13-s + (−0.255 − 0.0884i)14-s − 1.29·15-s + (−0.393 − 0.681i)16-s + (0.906 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821157 + 0.546219i\)
\(L(\frac12)\) \(\approx\) \(0.821157 + 0.546219i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.190 - 0.330i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.11 - 1.93i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.73 + 6.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.88 - 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.35 - 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (0.736 + 1.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.736 - 1.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.73 - 6.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.35 + 9.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.11 - 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−14.56454761009922008723439693998, −13.54874918285554397979028465060, −11.71978116591483988392727164869, −10.98317839457928490395678231713, −9.963985434852701662601982436292, −9.365255247777837370764012818202, −7.00073560932273874586189697192, −5.96738277904419845478407978227, −5.09779491066717041247569614310, −3.00255312100191105218792907301, 1.69057622283079173193073362078, 3.97646047461772695807843743915, 5.94049446538056836979371913842, 6.98263698841547473536537244743, 7.974674709273874811639719904103, 9.576703550189450260791544118253, 10.91420614036800776582806184832, 12.30857092311281099427437521250, 12.68174269319074604724993730510, 13.21773273479029602439484288605

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