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Properties

Label 1-304-304.83-r1-0-0
Degree $1$
Conductor $304$
Sign $0.337 + 0.941i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + 7-s + (0.5 + 0.866i)9-s + i·11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + i·27-s + (0.866 − 0.5i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)5-s + 7-s + (0.5 + 0.866i)9-s + i·11-s + (0.866 − 0.5i)13-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + i·27-s + (0.866 − 0.5i)29-s − 31-s + (−0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.337 + 0.941i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.029784020 + 2.133494604i\)
\(L(\frac12)\) \(\approx\) \(3.029784020 + 2.133494604i\)
\(L(1)\) \(\approx\) \(1.799214924 + 0.6443174081i\)
\(L(1)\) \(\approx\) \(1.799214924 + 0.6443174081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + T \)
11 \( 1 + iT \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−24.96837753089143245252713786959, −24.108651969383682102331268166143, −23.60555063325691813591008440384, −21.85384265003236012547689138312, −21.22687323804083999651525284095, −20.50167878793637829376422232384, −19.64791908310673057273877482707, −18.283181903785262453388042839199, −18.08237690586189553901302549791, −16.76675986065820889103684430706, −15.77435514279467779113989042526, −14.51827522993721838572378240181, −13.744604599861893743316745107278, −13.34355420471805822121378662596, −11.97825065030030061467628026850, −11.02058189311007217582093712996, −9.6243749245458955636065694266, −8.7497262570797901273092360126, −8.13685971614147074803919251725, −6.814928736710328394693799925911, −5.74242981179512401712056947234, −4.52902129051805828266953296772, −3.163146333320031429688916134, −1.884259217843808919874939412931, −1.07017186239897685107862285522, 1.66603770350533376172170568187, 2.40658766002047798897411377105, 3.82888493184641616600292849049, 4.839129094385960902206161493246, 6.06452683194976373637641807230, 7.38249791086308624059228613383, 8.370763749345599595258966483424, 9.267643423901613664635881916552, 10.42425489890416348346894044970, 10.85568981180212897925634044551, 12.52320209460686864914360619335, 13.53339950906174928468569336520, 14.39765786550037826981862268497, 14.979659782222770506136164959725, 15.924629868349766278486765182406, 17.34319412283414103150941618026, 17.95520537533046881549772084746, 18.916193408557546686713848153601, 20.219696548285163991584236879833, 20.71089953172469809505967719363, 21.592176638872641586499538131675, 22.32624300644268573445359245258, 23.51018243399468721955650829755, 24.70333544600504842254689897396, 25.34738791015712463098125820548

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