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Properties

Label 2-91-91.76-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.539 + 0.842i$
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  + (0.612 − 2.28i)2-s + (−1.04 + 0.601i)3-s + (−3.12 − 1.80i)4-s + (2.25 − 2.25i)5-s + (0.736 + 2.74i)6-s + (−2.60 + 0.457i)7-s + (−2.68 + 2.68i)8-s + (−0.777 + 1.34i)9-s + (−3.77 − 6.54i)10-s + (4.11 + 1.10i)11-s + 4.33·12-s + (3.48 + 0.920i)13-s + (−0.550 + 6.24i)14-s + (−0.992 + 3.70i)15-s + (0.897 + 1.55i)16-s + (0.721 − 1.25i)17-s + ⋯
L(s)  = 1  + (0.433 − 1.61i)2-s + (−0.601 + 0.347i)3-s + (−1.56 − 0.901i)4-s + (1.00 − 1.00i)5-s + (0.300 + 1.12i)6-s + (−0.984 + 0.172i)7-s + (−0.950 + 0.950i)8-s + (−0.259 + 0.448i)9-s + (−1.19 − 2.06i)10-s + (1.23 + 0.332i)11-s + 1.25·12-s + (0.966 + 0.255i)13-s + (−0.147 + 1.66i)14-s + (−0.256 + 0.956i)15-s + (0.224 + 0.388i)16-s + (0.175 − 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.539 + 0.842i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.539 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501363 - 0.916159i\)
\(L(\frac12)\) \(\approx\) \(0.501363 - 0.916159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.60 - 0.457i)T \)
13 \( 1 + (-3.48 - 0.920i)T \)
good2 \( 1 + (-0.612 + 2.28i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (1.04 - 0.601i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.25 + 2.25i)T - 5iT^{2} \)
11 \( 1 + (-4.11 - 1.10i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.721 + 1.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.649 - 2.42i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.52 - 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.34 - 2.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.76 - 3.76i)T - 31iT^{2} \)
37 \( 1 + (0.599 + 0.160i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.04 + 1.35i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.46 + 3.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.68 - 4.68i)T + 47iT^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (1.96 - 0.525i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.53 - 2.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 + 8.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.31 + 2.22i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.1 + 11.1i)T + 73iT^{2} \)
79 \( 1 - 2.01T + 79T^{2} \)
83 \( 1 + (-5.11 + 5.11i)T - 83iT^{2} \)
89 \( 1 + (1.81 - 6.76i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.62 - 6.06i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.49042677505553721488271825652, −12.43618950883037568334495217594, −11.78441780538436352970078290070, −10.54767167082299079766304904325, −9.657120498010588070462266808335, −8.944441765813305363434500387772, −6.16174800460343226880607690689, −5.08011072670274656121526948747, −3.69012710392454764877890818694, −1.65855353145960669693195440555, 3.64431159419840837727898099334, 5.86473618227276232776073735108, 6.30318323434263602208043281385, 6.94652968964390478085645905039, 8.701369038469230407193616686302, 9.892558837795063209165741244543, 11.33937199066200443183183963888, 12.81773868619627450425693244610, 13.75532944938722957673779029642, 14.41302293445040726859744700547

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