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Properties

Label 1-91-91.59-r0-0-0
Degree $1$
Conductor $91$
Sign $0.779 + 0.625i$
Analytic cond. $0.422602$
Root an. cond. $0.422602$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯
L(s)  = 1  + i·2-s + (0.5 − 0.866i)3-s − 4-s + (0.866 + 0.5i)5-s + (0.866 + 0.5i)6-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)15-s + 16-s + 17-s + (0.866 − 0.5i)18-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.779 + 0.625i$
Analytic conductor: \(0.422602\)
Root analytic conductor: \(0.422602\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 91,\ (0:\ ),\ 0.779 + 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.112534816 + 0.3911299016i\)
\(L(\frac12)\) \(\approx\) \(1.112534816 + 0.3911299016i\)
\(L(1)\) \(\approx\) \(1.156413946 + 0.3261377696i\)
\(L(1)\) \(\approx\) \(1.156413946 + 0.3261377696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + iT \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.133557892162467253772323918999, −29.34404558439472436113021556355, −27.96565411404875538913149614266, −27.60325219323456630340671673864, −26.206347375941238496610722937768, −25.37695292107608939342650291379, −23.88603959359456724491022059489, −22.237400471490481454908192741, −21.7094479314278484467045440937, −20.73543214795237476523145956968, −19.931324488345658577386011190235, −18.79376463880898416007120630382, −17.31374070350886971238514324418, −16.47038241305807360692028205802, −14.63800665301335706512090638775, −13.88453856479112139835976916850, −12.70934661965771298415299880271, −11.27824495308560913818817399333, −10.09619817142799019299818655887, −9.27197911243815642125971772914, −8.34063961243424624274279936082, −5.77394266120280268608953370886, −4.524514783920292227980988675094, −3.26634892294220613975896594681, −1.75830341764648968613489359920, 1.79428331052860771971284115703, 3.70711876351360264046503423456, 5.731447481157229711600216361736, 6.6327175674274969556143801245, 7.68981771894598556102375197413, 8.990821895152488903760020919548, 10.04527747971238999985139031744, 12.18130455106352443620670475528, 13.30551152423649127486178517117, 14.3479642233397157986987253519, 14.8583105844581390430183092569, 16.689279422142645868732274508469, 17.63664048050657853467642795171, 18.47315620065466373876004735078, 19.48458224622562189993594717496, 21.076855159753270406538826325227, 22.3742027474005171526206848548, 23.32478803013400650361668926186, 24.46679003901237006979859781149, 25.42841075781744173820194565634, 25.77629930740802395288805724881, 27.07247691419995534857983773900, 28.381523947790246233939539962185, 29.864373965940021778743060274094, 30.42662450524056725339077179878

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