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Properties

Label 2-91-13.3-c1-0-7
Degree $2$
Conductor $91$
Sign $-0.942 - 0.333i$
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  + (−1.11 − 1.92i)2-s + (−0.274 − 0.475i)3-s + (−1.46 + 2.53i)4-s − 4.22·5-s + (−0.610 + 1.05i)6-s + (0.5 − 0.866i)7-s + 2.06·8-s + (1.34 − 2.33i)9-s + (4.68 + 8.11i)10-s + (0.274 + 0.475i)11-s + 1.60·12-s + (−2.95 − 2.06i)13-s − 2.22·14-s + (1.15 + 2.00i)15-s + (0.640 + 1.10i)16-s + (1.18 − 2.06i)17-s + ⋯
L(s)  = 1  + (−0.784 − 1.35i)2-s + (−0.158 − 0.274i)3-s + (−0.732 + 1.26i)4-s − 1.88·5-s + (−0.249 + 0.431i)6-s + (0.188 − 0.327i)7-s + 0.728·8-s + (0.449 − 0.778i)9-s + (1.48 + 2.56i)10-s + (0.0828 + 0.143i)11-s + 0.464·12-s + (−0.820 − 0.571i)13-s − 0.593·14-s + (0.299 + 0.518i)15-s + (0.160 + 0.277i)16-s + (0.288 − 0.499i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.942 - 0.333i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.942 - 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0600101 + 0.349607i\)
\(L(\frac12)\) \(\approx\) \(0.0600101 + 0.349607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (2.95 + 2.06i)T \)
good2 \( 1 + (1.11 + 1.92i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.274 + 0.475i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 4.22T + 5T^{2} \)
11 \( 1 + (-0.274 - 0.475i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.18 + 2.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.90 + 5.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.79 - 3.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.14T + 31T^{2} \)
37 \( 1 + (-0.164 - 0.285i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.14 - 5.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.61 - 2.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.20T + 47T^{2} \)
53 \( 1 - 2.65T + 53T^{2} \)
59 \( 1 + (-0.903 + 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.304 - 0.527i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.18 + 8.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.90T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 5.73T + 83T^{2} \)
89 \( 1 + (-3.73 - 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.42 + 5.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.72098732622981601294405891412, −12.17586700928181408089458016103, −11.44040697046566554777701428811, −10.48478029384537397027395553833, −9.243786384027996485873986681033, −8.021045934767887545835457071775, −7.11264264174337572247168364437, −4.39589751016946934774997453766, −3.15821707461235387314895124712, −0.57882050756415191752513853396, 4.06718776761862603520325345627, 5.47578386847574600552693692732, 7.26723451669515982169042314209, 7.73834248384541340874100914753, 8.748213237749390345471798006750, 10.09876236597486755053135363478, 11.46588888505832269530310506439, 12.33714946226223589548149906026, 14.21442765294781831030764506338, 15.13816755149589831991045822230

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