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Properties

Label 2-91-91.59-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.0359 + 0.999i$
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  + (−0.876 − 0.876i)2-s + (1.92 − 1.11i)3-s − 0.463i·4-s + (−2.51 − 0.674i)5-s + (−2.66 − 0.713i)6-s + (2.20 + 1.46i)7-s + (−2.15 + 2.15i)8-s + (0.975 − 1.69i)9-s + (1.61 + 2.79i)10-s + (1.36 + 0.365i)11-s + (−0.515 − 0.893i)12-s + (0.445 − 3.57i)13-s + (−0.642 − 3.21i)14-s + (−5.60 + 1.50i)15-s + 2.85·16-s + 2.82·17-s + ⋯
L(s)  = 1  + (−0.619 − 0.619i)2-s + (1.11 − 0.642i)3-s − 0.231i·4-s + (−1.12 − 0.301i)5-s + (−1.08 − 0.291i)6-s + (0.831 + 0.554i)7-s + (−0.763 + 0.763i)8-s + (0.325 − 0.563i)9-s + (0.510 + 0.884i)10-s + (0.411 + 0.110i)11-s + (−0.148 − 0.257i)12-s + (0.123 − 0.992i)13-s + (−0.171 − 0.859i)14-s + (−1.44 + 0.387i)15-s + 0.714·16-s + 0.685·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0359 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0359 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0359 + 0.999i$
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: \((2,\ 91,\ (\ :1/2),\ -0.0359 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628629 - 0.651625i\)
\(L(\frac12)\) \(\approx\) \(0.628629 - 0.651625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.20 - 1.46i)T \)
13 \( 1 + (-0.445 + 3.57i)T \)
good2 \( 1 + (0.876 + 0.876i)T + 2iT^{2} \)
3 \( 1 + (-1.92 + 1.11i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.51 + 0.674i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.36 - 0.365i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (-1.61 - 6.04i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.01iT - 23T^{2} \)
29 \( 1 + (2.66 - 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 + 6.46i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.50 - 6.50i)T - 37iT^{2} \)
41 \( 1 + (2.51 + 9.40i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.850 + 0.490i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.594 + 2.21i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.52 - 4.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 + 5.39i)T + 59iT^{2} \)
61 \( 1 + (6.75 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.38 - 12.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.97 + 11.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.43 + 2.26i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.445 - 0.445i)T - 83iT^{2} \)
89 \( 1 + (-0.108 - 0.108i)T + 89iT^{2} \)
97 \( 1 + (2.87 + 0.771i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.00081307957853421987493129070, −12.42912153129946751170330270274, −11.79459759464194365247335348647, −10.57930024376442468687062211519, −9.177693251981806378310646019350, −8.229430780747191562871107183637, −7.70533394654794133143247878221, −5.47705414260797974243306536248, −3.38322353314134564666114055461, −1.65869008249956140726180464431, 3.34961767666041016120671014850, 4.33011226387163045435735933046, 6.95176814673962674769294741634, 7.80860528063651655421614046931, 8.667420234508003619843532974477, 9.531646370742689441329307035410, 11.10642645201179894152553253821, 12.04765774494946480490195451733, 13.75520522364225942223512450615, 14.63853881507329022012971403638

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