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Properties

Label 1-304-304.259-r0-0-0
Degree $1$
Conductor $304$
Sign $0.683 + 0.730i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(1.41177\)
Root analytic conductor: \(1.41177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (0:\ ),\ 0.683 + 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.011947591 + 0.4388651815i\)
\(L(\frac12)\) \(\approx\) \(1.011947591 + 0.4388651815i\)
\(L(1)\) \(\approx\) \(0.9758995216 + 0.1291603561i\)
\(L(1)\) \(\approx\) \(0.9758995216 + 0.1291603561i\)

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

−24.801223942333931217298094347171, −24.45848748622223005554836310554, −23.472301441819255479805470547668, −22.28317863587018670640044382817, −21.64737597052257248406990167388, −20.95616101366580012094319611056, −20.08326043510179694259733390421, −18.58390528743555515419063668197, −17.65623418372493836226078459813, −17.19381750127524991650499419881, −16.24799033181675340672360711569, −15.26562410130161483069717957294, −14.14091493716778352376957043708, −13.25175154252458634517418476292, −12.0061201492205891639052206578, −11.29548975917677767626299815921, −10.25753114167023203829969370235, −9.40082103459293690955973145784, −8.31616969848292591267808793982, −6.96066526109738611121124006234, −5.56426377093112512762358307095, −5.25659328741945346593090498757, −4.02638649671367993786174577013, −2.31241583951311950840938143136, −0.83817015751341874448185088727, 1.63119543164511409743247262914, 2.28489648955774409993112224677, 4.43809271571962222562151853224, 5.22309195315388810047840919612, 6.41893633450210023721664405521, 7.132555609293880268045406860210, 8.28548718146276883886497051737, 9.80246658941458208544488362156, 10.54206406826456333880498120464, 11.533998866963495113462383835896, 12.41640206702662659762965301165, 13.40652251411619987143675714299, 14.44846421269821346348803398826, 15.19526121899194274107792406075, 16.80892435739226898074726070104, 17.336818342657564310057804549364, 18.06799648951407050720829201667, 18.79182949934487938406468309667, 20.057700314173838688572489940019, 21.17584685140941055978038299405, 22.00451901806517639004304528639, 22.61500886019165516122396000733, 23.812185374999852596570022698469, 24.3915280258168570444057534808, 25.28845415058447200264639845360

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