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Properties

Label 2-91-91.9-c1-0-5
Degree $2$
Conductor $91$
Sign $-0.446 + 0.894i$
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  + (−0.777 − 1.34i)2-s − 0.489·3-s + (−0.208 + 0.361i)4-s + (0.595 − 1.03i)5-s + (0.380 + 0.658i)6-s + (0.337 − 2.62i)7-s − 2.46·8-s − 2.76·9-s − 1.85·10-s + 2.11·11-s + (0.102 − 0.176i)12-s + (2.86 + 2.19i)13-s + (−3.79 + 1.58i)14-s + (−0.291 + 0.504i)15-s + (2.33 + 4.03i)16-s + (0.453 − 0.784i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.952i)2-s − 0.282·3-s + (−0.104 + 0.180i)4-s + (0.266 − 0.461i)5-s + (0.155 + 0.268i)6-s + (0.127 − 0.991i)7-s − 0.870·8-s − 0.920·9-s − 0.585·10-s + 0.638·11-s + (0.0294 − 0.0510i)12-s + (0.793 + 0.608i)13-s + (−1.01 + 0.423i)14-s + (−0.0752 + 0.130i)15-s + (0.582 + 1.00i)16-s + (0.109 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.446 + 0.894i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.446 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379920 - 0.614422i\)
\(L(\frac12)\) \(\approx\) \(0.379920 - 0.614422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.337 + 2.62i)T \)
13 \( 1 + (-2.86 - 2.19i)T \)
good2 \( 1 + (0.777 + 1.34i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + 0.489T + 3T^{2} \)
5 \( 1 + (-0.595 + 1.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
17 \( 1 + (-0.453 + 0.784i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 6.69T + 19T^{2} \)
23 \( 1 + (1.79 + 3.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.25 - 7.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.49 + 4.32i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.768 - 1.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.71 + 4.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.59 + 2.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.41 - 2.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.12 + 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 + 3.74T + 67T^{2} \)
71 \( 1 + (-1.26 - 2.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.86 - 4.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + (-8.87 - 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.10 + 5.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.73597412054166275776152534714, −12.31169300297547596994304674733, −11.40501344819968613319340314032, −10.68071332953190586613059494521, −9.464955954852306698710736631762, −8.618624097799711629134223139248, −6.83009022983697630823688457272, −5.39680975432254820368914107497, −3.47056810512680249336675206530, −1.27123782086708242007570856389, 3.03989103486242207375010228791, 5.67236660392100763815191321723, 6.23218713955958868748894874254, 7.76205731357398955181813401179, 8.711973872079897816345154693488, 9.752504397559834595231411103159, 11.43396604264525097353784977865, 12.01004343992182019630057298011, 13.65944447167854448473283380483, 14.76432286932195200971627229481

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