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Properties

Label 2-91-91.31-c1-0-0
Degree $2$
Conductor $91$
Sign $0.0385 - 0.999i$
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  + (−1.58 − 0.423i)2-s + (−1.71 − 0.990i)3-s + (0.587 + 0.339i)4-s + (−0.742 + 2.77i)5-s + (2.29 + 2.29i)6-s + (1.81 + 1.92i)7-s + (1.52 + 1.52i)8-s + (0.462 + 0.801i)9-s + (2.34 − 4.06i)10-s + (−3.33 + 0.894i)11-s + (−0.671 − 1.16i)12-s + (−2.29 + 2.78i)13-s + (−2.04 − 3.81i)14-s + (4.02 − 4.02i)15-s + (−2.44 − 4.24i)16-s + (−3.22 + 5.58i)17-s + ⋯
L(s)  = 1  + (−1.11 − 0.299i)2-s + (−0.990 − 0.571i)3-s + (0.293 + 0.169i)4-s + (−0.332 + 1.24i)5-s + (0.936 + 0.936i)6-s + (0.684 + 0.729i)7-s + (0.540 + 0.540i)8-s + (0.154 + 0.267i)9-s + (0.742 − 1.28i)10-s + (−1.00 + 0.269i)11-s + (−0.193 − 0.335i)12-s + (−0.635 + 0.771i)13-s + (−0.546 − 1.02i)14-s + (1.03 − 1.03i)15-s + (−0.612 − 1.06i)16-s + (−0.782 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0385 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0385 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.0385 - 0.999i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.0385 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186985 + 0.179914i\)
\(L(\frac12)\) \(\approx\) \(0.186985 + 0.179914i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.81 - 1.92i)T \)
13 \( 1 + (2.29 - 2.78i)T \)
good2 \( 1 + (1.58 + 0.423i)T + (1.73 + i)T^{2} \)
3 \( 1 + (1.71 + 0.990i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.742 - 2.77i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.33 - 0.894i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.786 + 2.93i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.97 + 1.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.40T + 29T^{2} \)
31 \( 1 + (-3.24 + 0.868i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.00 - 3.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.03 - 3.03i)T + 41iT^{2} \)
43 \( 1 + 4.48iT - 43T^{2} \)
47 \( 1 + (5.51 + 1.47i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.72 + 9.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.766 - 2.86i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.03 - 1.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.88 - 7.04i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.31 + 8.31i)T - 71iT^{2} \)
73 \( 1 + (-1.20 - 4.51i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.543 + 0.942i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.01 + 2.01i)T + 83iT^{2} \)
89 \( 1 + (-4.79 - 1.28i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.20 - 3.20i)T + 97iT^{2} \)
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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

−14.57978091277577881519272756878, −13.02961968881660965442581303587, −11.66495527573975659357074638030, −11.13564964784143412717620331165, −10.28794016864142310552459158767, −8.829116422623790315979596918774, −7.61127585490560249247879319038, −6.58317445086193731157655750240, −5.03793447244144939790250479747, −2.23733696511419722037401966430, 0.50217844889111864523825724717, 4.53877031422089976007235046560, 5.29435697600451658573895785647, 7.43406146059490131283295428574, 8.216314214899960264459275619842, 9.450435478556155446987964155125, 10.49437709682851621950424318606, 11.31956579317497471708644571975, 12.65511156717211437792999915896, 13.68903990183899171990445073130

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