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Properties

Label 1-304-304.85-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.851 + 0.523i$
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  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)5-s + (0.5 + 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.342 − 0.939i)21-s + (−0.766 + 0.642i)23-s + (−0.173 − 0.984i)25-s + (−0.866 − 0.5i)27-s + (−0.342 + 0.939i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (−0.642 + 0.766i)5-s + (0.5 + 0.866i)7-s + (0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.984 − 0.173i)13-s + (0.766 − 0.642i)15-s + (−0.939 + 0.342i)17-s + (−0.342 − 0.939i)21-s + (−0.766 + 0.642i)23-s + (−0.173 − 0.984i)25-s + (−0.866 − 0.5i)27-s + (−0.342 + 0.939i)29-s + (−0.5 − 0.866i)31-s + (0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.851 + 0.523i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.851 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1092513878 + 0.3861581907i\)
\(L(\frac12)\) \(\approx\) \(0.1092513878 + 0.3861581907i\)
\(L(1)\) \(\approx\) \(0.5761105179 + 0.1781322976i\)
\(L(1)\) \(\approx\) \(0.5761105179 + 0.1781322976i\)

Euler product

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

−24.56284914638089881984466266444, −23.896719341175729040157865335530, −23.25006413453344481714595986511, −22.5617765522548544728626482165, −21.136802649376512739377817092439, −20.65904497255923400704763124515, −19.701754740219045448548252542955, −18.31047792520651862562944273108, −17.71278896164838079667808626323, −16.642139419329756774585332837235, −16.01206254417612286666690248072, −15.214347785621412301080967095888, −13.66729154581679589832776665064, −12.855683043673258065994670489179, −11.85435497966174118929305535541, −11.00223842306086401759923110348, −10.261923262237026589443924340604, −8.890001277922643031378689406, −7.77758141896495728869293690062, −6.86025474462435428875458380773, −5.51971788895154500904797438871, −4.5571912471234173938690542516, −3.894658090505133186925280317714, −1.71493893246089702590385645342, −0.297491274476636617518872739095, 1.744658514543940104591495824375, 3.17867128209038331882294526756, 4.51881046171019963709983166426, 5.6873825697803103321201334131, 6.43320045400127082171021262249, 7.6846263787269627841221915643, 8.5092108104590696384282566241, 10.09620341161793370647854994444, 11.286441444480360111727954683027, 11.313710080157599342043684851826, 12.66170476238097913010711771328, 13.58038663863113535347341541922, 15.03510769880986951998357541229, 15.64449906607501890326684287499, 16.46967927986249671964895392841, 17.949032650008005301505525676654, 18.261233686613779390538198876393, 19.03370942455448552102258810954, 20.31907173023294280829971510391, 21.64576380597573980736649833861, 21.99726810210493044091935326714, 23.13818822583258487287485027955, 23.76413751608954789724500403465, 24.51322790315197741265095759550, 25.792631199185602093443293525934

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