Properties

Label 2-91-7.4-c1-0-0
Degree $2$
Conductor $91$
Sign $0.936 - 0.349i$
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  + (−1.10 − 1.91i)2-s + (−1.23 + 2.14i)3-s + (−1.44 + 2.51i)4-s + (1.06 + 1.83i)5-s + 5.47·6-s + (2.63 + 0.272i)7-s + 1.98·8-s + (−1.56 − 2.70i)9-s + (2.34 − 4.06i)10-s + (−2.39 + 4.14i)11-s + (−3.58 − 6.21i)12-s + 13-s + (−2.39 − 5.34i)14-s − 5.25·15-s + (0.697 + 1.20i)16-s + (1.88 − 3.27i)17-s + ⋯
L(s)  = 1  + (−0.782 − 1.35i)2-s + (−0.714 + 1.23i)3-s + (−0.724 + 1.25i)4-s + (0.474 + 0.822i)5-s + 2.23·6-s + (0.994 + 0.102i)7-s + 0.703·8-s + (−0.520 − 0.901i)9-s + (0.742 − 1.28i)10-s + (−0.721 + 1.25i)11-s + (−1.03 − 1.79i)12-s + 0.277·13-s + (−0.638 − 1.42i)14-s − 1.35·15-s + (0.174 + 0.301i)16-s + (0.458 − 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577494 + 0.104217i\)
\(L(\frac12)\) \(\approx\) \(0.577494 + 0.104217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.63 - 0.272i)T \)
13 \( 1 - T \)
good2 \( 1 + (1.10 + 1.91i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.23 - 2.14i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.06 - 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.39 - 4.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.88 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 - 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.23 + 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + (-1.88 + 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (-3.55 - 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.19 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.39 - 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 + (3.85 - 6.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.58 - 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (1.83 + 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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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

−14.27337935298996128993016011834, −12.53078438802478121022677937612, −11.50884583225051438832565116121, −10.72464524775351696525340410892, −10.13881340757533615052462242536, −9.347926428421646456610868284682, −7.71290583146263995407545258438, −5.59642784637144022373725924927, −4.18767110669254397647783736591, −2.36060517314558161997451118530, 1.12783726141663635359438766510, 5.40016663764971863379182553645, 5.91328389419399910801102106711, 7.36181998675726873674072494321, 8.124011054761666687748373694087, 9.072974065715665633806741505527, 10.80061309196485148879829559611, 11.97565363803679616908655608875, 13.22899493257819188258363284456, 13.95499690662616336134306014104

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