Properties

Label 1-304-304.179-r0-0-0
Degree $1$
Conductor $304$
Sign $0.999 - 0.0330i$
Analytic cond. $1.41177$
Root an. cond. $1.41177$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.999 - 0.0330i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.999 - 0.0330i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.658117478 + 0.02736754357i\)
\(L(\frac12)\) \(\approx\) \(1.658117478 + 0.02736754357i\)
\(L(1)\) \(\approx\) \(1.370684422 - 0.05121653558i\)
\(L(1)\) \(\approx\) \(1.370684422 - 0.05121653558i\)

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

−25.22432489588249970895356788768, −24.415449282206169601227426693375, −23.80391379084602788253831270610, −22.60966140695875357516535931200, −21.39598098534829476162424939676, −20.883277703948023434344225375584, −19.965649169976605058167389134557, −19.25114798326689564629202903155, −18.24085850094605387389968663679, −16.984282454029188605452472010781, −15.94316520578482833021136372418, −15.38897450685749156267614271067, −14.357469375158336925282577516234, −13.53340976735891431143199711864, −12.419480039766621521962801835401, −11.1580778029597893685663494469, −10.57818276948555516434983197726, −9.00256577564526329217478094810, −8.26226193544593821092906960778, −7.86695897453857629477333979848, −6.08620439907797318231562167499, −4.67424360842209317568712633495, −3.99908487274460559695015635207, −2.82694440992677325210044810674, −1.26609937160016274050044963239, 1.42806572618333999370289658856, 2.58179767435972632377288653572, 3.82930592667673492994427049364, 4.74036559492567798180488071290, 6.58078125044898069959576808515, 7.389498151969171849068003164450, 8.183909522945752272014178440690, 9.09338950521050211046582128157, 10.40433307986318573909019344636, 11.609276419782717829625422007563, 12.18276980517851562744763092588, 13.634864348988892917600061302021, 14.21983027275196210390953145270, 15.288153689631662727698965094252, 15.75504719544552426480374974212, 17.47844867939691005613812546250, 18.2462010233113721899476179719, 18.942961479746232491628043319095, 20.05163513603811801075495611574, 20.53464062555134935002880424703, 21.5802062571797331375671482335, 22.896498723620148722547351523362, 23.637939133089488034013864535129, 24.34526996665123663048002322864, 25.41918897215654124069428139672

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