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Properties

Label 1-319-319.35-r1-0-0
Degree $1$
Conductor $319$
Sign $-0.282 + 0.959i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (−0.393 + 0.919i)2-s + (0.691 − 0.722i)3-s + (−0.691 − 0.722i)4-s + (0.753 + 0.657i)5-s + (0.393 + 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.936 − 0.351i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s − 12-s + (0.963 − 0.266i)13-s + (−0.858 − 0.512i)14-s + (0.995 − 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + (0.936 + 0.351i)18-s + ⋯
L(s)  = 1  + (−0.393 + 0.919i)2-s + (0.691 − 0.722i)3-s + (−0.691 − 0.722i)4-s + (0.753 + 0.657i)5-s + (0.393 + 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.936 − 0.351i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s − 12-s + (0.963 − 0.266i)13-s + (−0.858 − 0.512i)14-s + (0.995 − 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + (0.936 + 0.351i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.215306675 + 1.623945674i\)
\(L(\frac12)\) \(\approx\) \(1.215306675 + 1.623945674i\)
\(L(1)\) \(\approx\) \(1.075867298 + 0.5360408484i\)
\(L(1)\) \(\approx\) \(1.075867298 + 0.5360408484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.393 + 0.919i)T \)
3 \( 1 + (0.691 - 0.722i)T \)
5 \( 1 + (0.753 + 0.657i)T \)
7 \( 1 + (-0.134 + 0.990i)T \)
13 \( 1 + (0.963 - 0.266i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.134 + 0.990i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.393 - 0.919i)T \)
37 \( 1 + (0.550 - 0.834i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.550 + 0.834i)T \)
53 \( 1 + (-0.393 + 0.919i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.983 - 0.178i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.0448 + 0.998i)T \)
73 \( 1 + (-0.995 + 0.0896i)T \)
79 \( 1 + (-0.0448 - 0.998i)T \)
83 \( 1 + (-0.473 + 0.880i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
97 \( 1 + (-0.983 - 0.178i)T \)
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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

−24.98983206893273959687818388495, −23.70290817426899943913517458412, −22.487122322627253909036774291488, −21.71830075729900224643325372179, −20.83106546247139634880953049184, −20.30037747335539465279458735929, −19.71216743466135543717602593347, −18.45538857795089021641361439410, −17.48199136604139344794241435527, −16.598343634782327680697495685133, −15.877495056886645741814417157, −14.212362986493840608243109079757, −13.54220049762854698703290636987, −12.998096033032114173498715630010, −11.43148721372772450120742544006, −10.54476347146024958740754574667, −9.78893184878775269241091012569, −8.919361499794702341802001895056, −8.26124392346207306647067816410, −6.78519799291964621860142416575, −4.923412477965179843783290705857, −4.263331266446305265553294368320, −3.09393020628284820422516556974, −1.946935936426715327025110051, −0.654139137798256429965466710780, 1.35922508658359381460982393461, 2.414679596168811102713912926047, 3.82990443058791371734724297553, 5.87241483366613851380249065319, 6.045868653678749082499351646388, 7.28191526772871079260406149910, 8.27201275945763544900108845351, 9.09420224902334375977687919145, 9.91554842797905486422329365763, 11.22108010801244105187275658997, 12.75692071499204596644411305157, 13.50291973977043460118934913308, 14.369596716114554056604299456573, 15.138216772841547046752761057463, 15.91986875140583555952613797743, 17.42399356830472310182514401260, 18.01336802072365946571070764042, 18.73500764064785430101351027375, 19.35251380352091276736316680682, 20.670212849889658262154968472227, 21.77437486917443275018874561952, 22.71962207512354801027786005286, 23.63736754912004790533048972178, 24.7202006009805827197450683066, 25.16735237788603669782031240654

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