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Properties

Label 1-304-304.11-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: $\chi_{304} (11, \cdot )$
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

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

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