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Properties

Label 1-304-304.203-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.537 - 0.843i$
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  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (0.642 + 0.766i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.984 − 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)27-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (0.642 + 0.766i)13-s + (−0.939 − 0.342i)15-s + (0.173 − 0.984i)17-s + (−0.984 − 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (−0.866 − 0.5i)27-s + (−0.984 + 0.173i)29-s + (−0.5 − 0.866i)31-s + (0.939 − 0.342i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6382922112 - 1.164354005i\)
\(L(\frac12)\) \(\approx\) \(0.6382922112 - 1.164354005i\)
\(L(1)\) \(\approx\) \(0.9836495314 - 0.6331126051i\)
\(L(1)\) \(\approx\) \(0.9836495314 - 0.6331126051i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.642 - 0.766i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.173 - 0.984i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (0.984 + 0.173i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−25.75772060517645932450880795279, −25.05841358010348162160513911146, −23.84323593311085169501813385893, −22.46858331883065457034362305451, −22.18930608979120075876266884362, −21.30834239690859468855025563016, −20.13095780420664291515762168795, −19.29781917344605062565175150188, −18.70967320271722072966879530581, −17.48192210169290392567184972991, −16.177233144901650154029494668, −15.51921858144102405632518100949, −14.7480998194541760423742044151, −13.96891715513472503677401936260, −12.74827342044257661756290243835, −11.50804018149694683677224537948, −10.64962505615768455918954293587, −9.71072219730424313273927356535, −8.6967003277673095400140829230, −7.872683366856392419534414111699, −6.41984892838004870016990991245, −5.54342135441961641646706851026, −3.78230574156490466112841125685, −3.37000458818555497855882394339, −2.0916152568359618832492102392, 0.82772757800812443613572193830, 1.96547541849632955709443334858, 3.64025461715685250744695424475, 4.3218305162631068443737683344, 6.0319473158142403057118649435, 7.090741997491548097045752447935, 7.84987345279606243099235639620, 9.09617448745679106738199622759, 9.57939582573957542889168351728, 11.3203704986279663471329428032, 12.20119523262651067152000676031, 13.08168399313315913973499484410, 13.83899678528943314761507292617, 14.72005969820883329116751012694, 16.10672113626103390301382255930, 16.71969910483278620274325399486, 17.82785444598743819134741090467, 18.85069632926085777093219337138, 19.83419565247509477093108387534, 20.23525785593859005974433517770, 21.08035836106896997748299579173, 22.629950771278932057551941225858, 23.384087269248149405729749905263, 24.20435368620234823255288558490, 24.91456703936259307901429109077

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