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Properties

Label 2-91-91.23-c1-0-3
Degree $2$
Conductor $91$
Sign $0.372 - 0.927i$
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  + 1.73i·2-s + (0.5 − 0.866i)3-s − 0.999·4-s + (1.5 + 0.866i)5-s + (1.49 + 0.866i)6-s + (−2 − 1.73i)7-s + 1.73i·8-s + (1 + 1.73i)9-s + (−1.49 + 2.59i)10-s + (−4.5 − 2.59i)11-s + (−0.499 + 0.866i)12-s + (−1 − 3.46i)13-s + (2.99 − 3.46i)14-s + (1.5 − 0.866i)15-s − 5·16-s + 6·17-s + ⋯
L(s)  = 1  + 1.22i·2-s + (0.288 − 0.499i)3-s − 0.499·4-s + (0.670 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.755 − 0.654i)7-s + 0.612i·8-s + (0.333 + 0.577i)9-s + (−0.474 + 0.821i)10-s + (−1.35 − 0.783i)11-s + (−0.144 + 0.249i)12-s + (−0.277 − 0.960i)13-s + (0.801 − 0.925i)14-s + (0.387 − 0.223i)15-s − 1.25·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.372 - 0.927i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.372 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921335 + 0.622678i\)
\(L(\frac12)\) \(\approx\) \(0.921335 + 0.622678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.5 - 4.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 3.46iT - 59T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 - 0.866i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.20160350118561470449306431338, −13.55440261211125451535870249322, −12.70550912050822825842063304307, −10.73801558792837463651405583694, −9.970315481442408795627502753107, −8.073405316114595917085535553671, −7.58777677589498127478623729835, −6.30089923852536836045224263643, −5.32886254199497551754572328547, −2.80303467746832941482307451990, 2.18642467623826182589463232876, 3.61133977084879571321264428147, 5.29003435010259500926597715634, 6.98248780947366634934788759535, 8.996556265439986671266390640436, 9.803648140997261775495606943072, 10.32864071308234210480674182498, 11.98637722059681998716422776831, 12.59200406575555224235853639917, 13.48443661857895039273340476120

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