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Properties

Label 1-304-304.45-r0-0-0
Degree $1$
Conductor $304$
Sign $0.941 - 0.337i$
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

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

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.941 - 0.337i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.941 - 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001329803 - 0.1738200413i\)
\(L(\frac12)\) \(\approx\) \(1.001329803 - 0.1738200413i\)
\(L(1)\) \(\approx\) \(0.9070356242 - 0.09398065067i\)
\(L(1)\) \(\approx\) \(0.9070356242 - 0.09398065067i\)

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

−25.473609199642245376433802687203, −24.49549615251342444526591384984, −23.32766587885394123270674823169, −22.67283902834857026747238404524, −21.908628735303545469625029368604, −20.88088674103234532363393695748, −20.32164663227153531938783963008, −18.871503439159086524219861481495, −17.906791440210065322453554841437, −17.20380908720080246191030995291, −16.196465931904730441059762987275, −15.744969378225601006400675837680, −14.33710036592239905247440927929, −13.15694026996937337413656039336, −12.5125669584347063535120747736, −11.42370615059057933197458318996, −10.21631370732667769733316662488, −9.63314097886165239791415744499, −8.74377240088991382051376361015, −6.78236208777843738055779506303, −6.33064826285364799625518085902, −5.068793374187786798814098641431, −4.270178512258256135333914387859, −2.705701553479174221007786433618, −1.07611513164130075458211897875, 0.993771550134941014831152672612, 2.48060580518542844870191164367, 3.71078490741295473354630360257, 5.47620729706322260990078187937, 6.15065024939347546687036821631, 6.77869543894153547304735364049, 8.19242875890524219559626676638, 9.46934516682600505702230836485, 10.56794327689249546477341206878, 11.10169598659108301982921109435, 12.46623302487382149406529467929, 13.32885938091298263977121695753, 13.837074624619394083177004478632, 15.4522064034203438520968193586, 16.25406536165854652966457108490, 17.286917386146589857949030456496, 17.90015170348292162635656960091, 18.97288559852517363092069545184, 19.4587970607161225842262666935, 21.13914616293113535702804457474, 21.79473352714413429594313181633, 22.673280639177305092792598250948, 23.28225046829254248746757468487, 24.424835662651078882282623311999, 25.22602085830153002914401640186

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