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Properties

Label 2-91-91.89-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.962 + 0.272i$
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  + (−0.984 − 0.984i)2-s + (−1.25 − 0.724i)3-s − 0.0619i·4-s + (−0.172 − 0.643i)5-s + (0.521 + 1.94i)6-s + (−2.46 + 0.960i)7-s + (−2.02 + 2.02i)8-s + (−0.450 − 0.780i)9-s + (−0.463 + 0.802i)10-s + (−1.24 − 4.65i)11-s + (−0.0448 + 0.0776i)12-s + (3.60 + 0.0282i)13-s + (3.37 + 1.48i)14-s + (−0.249 + 0.931i)15-s + 3.87·16-s + 0.467·17-s + ⋯
L(s)  = 1  + (−0.696 − 0.696i)2-s + (−0.724 − 0.418i)3-s − 0.0309i·4-s + (−0.0770 − 0.287i)5-s + (0.213 + 0.795i)6-s + (−0.931 + 0.363i)7-s + (−0.717 + 0.717i)8-s + (−0.150 − 0.260i)9-s + (−0.146 + 0.253i)10-s + (−0.376 − 1.40i)11-s + (−0.0129 + 0.0224i)12-s + (0.999 + 0.00782i)13-s + (0.901 + 0.395i)14-s + (−0.0644 + 0.240i)15-s + 0.968·16-s + 0.113·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.962 + 0.272i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0564705 - 0.407171i\)
\(L(\frac12)\) \(\approx\) \(0.0564705 - 0.407171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.46 - 0.960i)T \)
13 \( 1 + (-3.60 - 0.0282i)T \)
good2 \( 1 + (0.984 + 0.984i)T + 2iT^{2} \)
3 \( 1 + (1.25 + 0.724i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.172 + 0.643i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.24 + 4.65i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.467T + 17T^{2} \)
19 \( 1 + (-3.26 - 0.873i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.95iT - 23T^{2} \)
29 \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.10 + 1.09i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.38 - 2.38i)T - 37iT^{2} \)
41 \( 1 + (-3.68 - 0.986i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.64 - 2.58i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.20 - 3.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.33 - 4.33i)T + 59iT^{2} \)
61 \( 1 + (-4.21 + 2.43i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.03 - 2.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.19 + 0.857i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.0301 - 0.112i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.194 + 0.337i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (-6.83 - 6.83i)T + 89iT^{2} \)
97 \( 1 + (4.61 + 17.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.27768266245316109064076270772, −12.27317777582661924091413132323, −11.35514876347539828806416954029, −10.54594243528466436964729771954, −9.228507683102188813100261763998, −8.407620697697599736266317667657, −6.34761955190753202711115555274, −5.66603505115740834464194385750, −3.11293225848952920042590814833, −0.68484255064352427069111382166, 3.55422060863412940283843377287, 5.42480643743911343545089810350, 6.79772597347719281572412439298, 7.62042038146658830272843520439, 9.179049947770543641124921342946, 10.07191477716812111318314517019, 11.15993416931236022446562656214, 12.45417817395701758772838411414, 13.42657467558342557037786585502, 15.03279395651624849852675080445

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