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Properties

Label 1-119-119.65-r1-0-0
Degree $1$
Conductor $119$
Sign $0.452 + 0.892i$
Analytic cond. $12.7883$
Root an. cond. $12.7883$
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  + (−0.965 − 0.258i)2-s + (0.793 − 0.608i)3-s + (0.866 + 0.5i)4-s + (−0.991 + 0.130i)5-s + (−0.923 + 0.382i)6-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (0.991 + 0.130i)10-s + (−0.130 + 0.991i)11-s + (0.991 − 0.130i)12-s + i·13-s + (−0.707 + 0.707i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.965 − 0.258i)19-s + (−0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.793 − 0.608i)3-s + (0.866 + 0.5i)4-s + (−0.991 + 0.130i)5-s + (−0.923 + 0.382i)6-s + (−0.707 − 0.707i)8-s + (0.258 − 0.965i)9-s + (0.991 + 0.130i)10-s + (−0.130 + 0.991i)11-s + (0.991 − 0.130i)12-s + i·13-s + (−0.707 + 0.707i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + (−0.965 − 0.258i)19-s + (−0.923 − 0.382i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.452 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.452 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.452 + 0.892i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ 0.452 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7026094024 + 0.4316273036i\)
\(L(\frac12)\) \(\approx\) \(0.7026094024 + 0.4316273036i\)
\(L(1)\) \(\approx\) \(0.7195377283 + 0.02343495389i\)
\(L(1)\) \(\approx\) \(0.7195377283 + 0.02343495389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.793 - 0.608i)T \)
5 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (0.793 + 0.608i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.793 + 0.608i)T \)
37 \( 1 + (0.130 + 0.991i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (-0.608 + 0.793i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.923 - 0.382i)T \)
73 \( 1 + (0.608 + 0.793i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−28.24727428828413989354374382068, −27.36064243105797752484247406131, −26.897178013845682742663024484320, −25.87749692006773968296489394911, −24.86221860399025377649403258454, −23.990628794629867014429065674802, −22.64674519201349205062582216833, −21.09871046497281740438222643203, −20.32910992499776016310218955857, −19.3031827758266963251549996609, −18.763990206185587263223474671373, −17.083247279176549109198257380524, −16.118453549341315286524806188121, −15.34448915960507444921090401775, −14.488940110126021378200355582250, −12.82147397100128142645780548824, −11.17416755119684733067111453048, −10.47162377498500023210094685748, −9.0094218629160421554779280071, −8.275279437191863714607528178297, −7.38536958214889039131970290984, −5.59932387980504528657054052279, −3.87647237797388976297290429420, −2.56944492582886882187067610685, −0.43502786334169966517279069261, 1.44677226225344095643210226880, 2.794363981533293869980366097735, 4.13777421270888844074948117212, 6.79179239292047420359516327332, 7.39997757135530424119728349227, 8.55099966124269873063091515678, 9.43574186201292199487142176833, 10.93623790038756965967586671580, 12.05567422194037073917020287827, 12.91777727641201874643692371290, 14.67784644323290894341428417269, 15.45024468104503878398056390774, 16.736891642287153443922605551346, 18.01759627829912799573354890383, 18.90608630598918778350376921101, 19.6479411161227820624084571019, 20.414637352900376632271843216219, 21.52938344322086757691155076327, 23.31856519130944566736095615872, 24.10485945466005966283152406965, 25.368976511491530805274965572715, 26.02930345594746351433613140285, 26.99348124874957108056527239644, 27.91237228885080628499606465808, 29.0261818077574241104370158802

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