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Properties

Label 1-304-304.253-r0-0-0
Degree $1$
Conductor $304$
Sign $-0.368 + 0.929i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ -0.368 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6558138442 + 0.9655770726i\)
\(L(\frac12)\) \(\approx\) \(0.6558138442 + 0.9655770726i\)
\(L(1)\) \(\approx\) \(0.8737496039 + 0.5277126383i\)
\(L(1)\) \(\approx\) \(0.8737496039 + 0.5277126383i\)

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

−24.88467625943684272984981526677, −24.06883928789892282727677959610, −23.50036368619723292162960510244, −22.49667581823243369865184190621, −21.46960627414831850557366116316, −20.44670602917924469458786251111, −19.703403104240414775297340061939, −18.637778350835356768388616039831, −17.54142351440121165042541932080, −16.993843200711398633764035375548, −16.37792285712550058138758269883, −14.85498565082840625809718607917, −13.64284055669425946661094570385, −13.12555553919396659224257044579, −12.139504116122267582119314330246, −11.12803354122726619035383007079, −10.30566259418829836797413144368, −8.77146100447556107736693321765, −8.037757981075686921001976618088, −6.83472194491101600035147252328, −5.83210986424440085472318288021, −4.91341649824255364452978690293, −3.65223468571949303082589233166, −1.662875200487151437822857379087, −0.93526077279232939128927972497, 1.74034897096722977565615809738, 3.148113715141874780646748335, 4.31649242606321964473539107306, 5.48158832167564565079833276177, 6.334415458890799372964058435207, 7.358062219016950436017746615982, 9.17616850455704660914835989969, 9.46788361010258019814226622651, 11.00471059489827231020982614025, 11.35552122236513837613480042305, 12.381444875879124320648188043292, 13.943912703255061543093969043266, 14.77099284207709672194225820610, 15.43160801357556355920973965856, 16.535615983376876833849431486797, 17.48459037946534806569557424805, 18.27575415740183585036658909656, 19.02458778235474540968396884457, 20.55043738451922998937299833378, 21.26132133312481352174954566740, 22.11589945937620343981196558994, 22.65881605461451845970635175771, 23.59304712278434383327340066324, 24.83747631948543142099123035061, 25.66783841425993329585494459665

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