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Properties

Label 2-91-91.5-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.730 - 0.682i$
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  + (0.423 + 1.58i)2-s + (−1.71 + 0.990i)3-s + (−0.587 + 0.339i)4-s + (−2.77 + 0.742i)5-s + (−2.29 − 2.29i)6-s + (1.92 + 1.81i)7-s + (1.52 + 1.52i)8-s + (0.462 − 0.801i)9-s + (−2.34 − 4.06i)10-s + (0.894 − 3.33i)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  + (0.299 + 1.11i)2-s + (−0.990 + 0.571i)3-s + (−0.293 + 0.169i)4-s + (−1.24 + 0.332i)5-s + (−0.936 − 0.936i)6-s + (0.729 + 0.684i)7-s + (0.540 + 0.540i)8-s + (0.154 − 0.267i)9-s + (−0.742 − 1.28i)10-s + (0.269 − 1.00i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.730 - 0.682i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.730 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304472 + 0.771491i\)
\(L(\frac12)\) \(\approx\) \(0.304472 + 0.771491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−15.04276645426449613004246929540, −13.81477660480904799896152045658, −12.08598462837375797203233063947, −11.23811082064404207772693835588, −10.63732296576525243531470952304, −8.403465184148286777153389282877, −7.75221355158063088336178378325, −6.05703553586617263981447486756, −5.42680999524761518805342151622, −3.93067302731984187601894596567, 1.24189235928451201260486154082, 3.77933818344139240339711422410, 4.88987331756444268775912689913, 6.96792199497708846396116618233, 7.74060471982599193297121720089, 9.693338628223814491218769644194, 11.11455646285982002528836516127, 11.76410289169975986091005504247, 12.03828282386098943184004652878, 13.25723889176947469407894006635

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