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Properties

Label 1-241-241.235-r0-0-0
Degree $1$
Conductor $241$
Sign $0.674 + 0.738i$
Analytic cond. $1.11919$
Root an. cond. $1.11919$
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  − 2-s + (0.809 + 0.587i)3-s + 4-s + (0.809 − 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s − 8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + i·11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s + (−0.951 − 0.309i)14-s + 15-s + 16-s + (−0.951 − 0.309i)17-s + ⋯
L(s)  = 1  − 2-s + (0.809 + 0.587i)3-s + 4-s + (0.809 − 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.951 + 0.309i)7-s − 8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + i·11-s + (0.809 + 0.587i)12-s + (−0.587 + 0.809i)13-s + (−0.951 − 0.309i)14-s + 15-s + 16-s + (−0.951 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(241\)
Sign: $0.674 + 0.738i$
Analytic conductor: \(1.11919\)
Root analytic conductor: \(1.11919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{241} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 241,\ (0:\ ),\ 0.674 + 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.132627205 + 0.4992946736i\)
\(L(\frac12)\) \(\approx\) \(1.132627205 + 0.4992946736i\)
\(L(1)\) \(\approx\) \(1.042782950 + 0.2542236183i\)
\(L(1)\) \(\approx\) \(1.042782950 + 0.2542236183i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−26.05109071870718650785724113241, −25.32736476412443322687609592597, −24.34125328671577881786865376393, −23.947628315624829527592777983251, −22.02307825762466741345632826032, −21.19579907878619438645245664340, −20.25907445100632014928843228458, −19.45579954056872443301536970201, −18.52119136794660644799938089227, −17.68363529782547279077567490294, −17.23630482412886954308530040092, −15.55855307497880964856872353387, −14.73647299919379700188914652549, −13.83313837170004449004946226716, −12.80090483107640087045536748718, −11.269544274556778570918704472681, −10.63278199209384609142961047953, −9.321929769012834066991124263403, −8.586125304067325365888904974382, −7.49749593190060753165646327389, −6.766540680658379545010214578891, −5.46225401738348799909566599063, −3.28094028119744795453906217047, −2.32252997263203818202643263500, −1.228905086103192224707367196353, 1.87884073310628206032200440118, 2.25837977253512312183980774560, 4.282490953379306425836865152693, 5.34545112662583482284661203108, 6.94235694545814082733007253652, 8.061811241207286476016129600735, 8.9711702823521909749499458806, 9.59693492207146364808299186778, 10.52682831848913587155032415009, 11.73633531951498528890252706254, 12.92886938992671126234904976435, 14.37487337280350656174043010068, 14.931743450966048163865111588754, 16.12541684017275116165626649302, 16.9969830618348537795846121347, 17.84388102393887357539610657925, 18.809347203134174917266711000101, 19.95347707900943320529630733347, 20.75909117408947575717792897891, 21.13259196822379373245625796198, 22.25458558242956465464652038160, 24.11445366033647494363053287123, 24.83010517343070514066324050331, 25.35213320102946707237825057157, 26.43730289328390875814202916152

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