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Properties

Label 1-91-91.54-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: $\chi_{91} (54, \cdot )$
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

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

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