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Properties

Label 2-91-91.73-c1-0-6
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: $\chi_{91} (73, \cdot )$
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

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

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