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Properties

Label 2-91-7.4-c1-0-3
Degree $2$
Conductor $91$
Sign $0.386 - 0.922i$
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  + (1.30 + 2.26i)2-s + (1.11 − 1.93i)3-s + (−2.42 + 4.20i)4-s + (−1.11 − 1.93i)5-s + 5.85·6-s + (−2 + 1.73i)7-s − 7.47·8-s + (−1 − 1.73i)9-s + (2.92 − 5.06i)10-s + (1.5 − 2.59i)11-s + (5.42 + 9.39i)12-s − 13-s + (−6.54 − 2.26i)14-s − 5.00·15-s + (−4.92 − 8.53i)16-s + (−0.736 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.925 + 1.60i)2-s + (0.645 − 1.11i)3-s + (−1.21 + 2.10i)4-s + (−0.499 − 0.866i)5-s + 2.38·6-s + (−0.755 + 0.654i)7-s − 2.64·8-s + (−0.333 − 0.577i)9-s + (0.925 − 1.60i)10-s + (0.452 − 0.783i)11-s + (1.56 + 2.71i)12-s − 0.277·13-s + (−1.74 − 0.605i)14-s − 1.29·15-s + (−1.23 − 2.13i)16-s + (−0.178 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.386 - 0.922i$
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: \((2,\ 91,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23124 + 0.819000i\)
\(L(\frac12)\) \(\approx\) \(1.23124 + 0.819000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + T \)
good2 \( 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.736 - 1.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.35 + 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.73 - 6.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.73 + 6.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.736 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (-1.35 + 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.35 - 2.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.11 + 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−14.20626976383485173222061194435, −13.30654486884255639490446195010, −12.76409423261741952593782406923, −11.91510771744855902170491470600, −8.946273652550477929618178825074, −8.458333016419311493913696830101, −7.28707580737416211507804133607, −6.37522657177174766914774324039, −5.05436121462349775984850677814, −3.35327604878179764201177913968, 2.82081953332938905818375982573, 3.77734419038093641113834944118, 4.64406321105409139521068003200, 6.77337655772861857255230814779, 9.025745998523202722389118705428, 10.19511879307132675294626325601, 10.41507271308888056682771179061, 11.71423185081363697482551012281, 12.75344849804855770208887521365, 13.89616788215796578725131254382

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