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Properties

Label 2-91-91.4-c1-0-2
Degree $2$
Conductor $91$
Sign $0.985 - 0.169i$
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  + 0.499i·2-s + (−0.424 − 0.735i)3-s + 1.75·4-s + (−0.902 + 0.521i)5-s + (0.367 − 0.212i)6-s + (2.63 − 0.239i)7-s + 1.87i·8-s + (1.13 − 1.97i)9-s + (−0.260 − 0.451i)10-s + (−3.43 + 1.98i)11-s + (−0.743 − 1.28i)12-s + (−3.57 + 0.468i)13-s + (0.119 + 1.31i)14-s + (0.767 + 0.442i)15-s + 2.56·16-s + 0.142·17-s + ⋯
L(s)  = 1  + 0.353i·2-s + (−0.245 − 0.424i)3-s + 0.875·4-s + (−0.403 + 0.233i)5-s + (0.150 − 0.0867i)6-s + (0.995 − 0.0904i)7-s + 0.662i·8-s + (0.379 − 0.657i)9-s + (−0.0824 − 0.142i)10-s + (−1.03 + 0.598i)11-s + (−0.214 − 0.371i)12-s + (−0.991 + 0.129i)13-s + (0.0319 + 0.352i)14-s + (0.198 + 0.114i)15-s + 0.640·16-s + 0.0344·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05362 + 0.0898873i\)
\(L(\frac12)\) \(\approx\) \(1.05362 + 0.0898873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 + 0.239i)T \)
13 \( 1 + (3.57 - 0.468i)T \)
good2 \( 1 - 0.499iT - 2T^{2} \)
3 \( 1 + (0.424 + 0.735i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.902 - 0.521i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.43 - 1.98i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.142T + 17T^{2} \)
19 \( 1 + (4.77 + 2.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (-4.19 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.46 - 1.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.843iT - 37T^{2} \)
41 \( 1 + (-10.4 - 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.94 - 2.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.139 + 0.242i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 + (-2.93 + 5.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.45 - 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.20 - 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.72 - 3.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.87iT - 83T^{2} \)
89 \( 1 - 1.74iT - 89T^{2} \)
97 \( 1 + (-2.34 + 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.46419863401419939318891532826, −12.86745599132455115972728932611, −11.92118513877710938426060781467, −11.12561083645861939869369257616, −9.932774443759250510199029162231, −7.993788606721441201525239310965, −7.38734621365964983066744028852, −6.20812773878653709232275452849, −4.62307160903893714184466982493, −2.28239241536009428385497676937, 2.30329242869913836253238432207, 4.35783861857568312560413271774, 5.65962833562230373513083547253, 7.46281078387609720412518527086, 8.282259718285237248874925104114, 10.27413378159239470505008286481, 10.71566159826001908129829829738, 11.84695323823485242008428052738, 12.67689958778295906765308660111, 14.16604142310747212667414673682

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