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Properties

Label 1-2347-2347.34-r1-0-0
Degree $1$
Conductor $2347$
Sign $-0.0476 + 0.998i$
Analytic cond. $252.220$
Root an. cond. $252.220$
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  + (0.999 + 0.0401i)2-s + (0.817 − 0.575i)3-s + (0.996 + 0.0802i)4-s + (0.216 − 0.976i)5-s + (0.840 − 0.542i)6-s + (−0.900 + 0.434i)7-s + (0.992 + 0.120i)8-s + (0.337 − 0.941i)9-s + (0.255 − 0.966i)10-s + (−0.478 − 0.877i)11-s + (0.861 − 0.508i)12-s + (−0.460 + 0.887i)13-s + (−0.917 + 0.398i)14-s + (−0.384 − 0.922i)15-s + (0.987 + 0.160i)16-s + (−0.572 + 0.820i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0401i)2-s + (0.817 − 0.575i)3-s + (0.996 + 0.0802i)4-s + (0.216 − 0.976i)5-s + (0.840 − 0.542i)6-s + (−0.900 + 0.434i)7-s + (0.992 + 0.120i)8-s + (0.337 − 0.941i)9-s + (0.255 − 0.966i)10-s + (−0.478 − 0.877i)11-s + (0.861 − 0.508i)12-s + (−0.460 + 0.887i)13-s + (−0.917 + 0.398i)14-s + (−0.384 − 0.922i)15-s + (0.987 + 0.160i)16-s + (−0.572 + 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0476 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0476 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2347\)
Sign: $-0.0476 + 0.998i$
Analytic conductor: \(252.220\)
Root analytic conductor: \(252.220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2347} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2347,\ (1:\ ),\ -0.0476 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.459084782 + 1.530317195i\)
\(L(\frac12)\) \(\approx\) \(1.459084782 + 1.530317195i\)
\(L(1)\) \(\approx\) \(1.973824647 - 0.3427816331i\)
\(L(1)\) \(\approx\) \(1.973824647 - 0.3427816331i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2347 \( 1 \)
good2 \( 1 + (0.999 + 0.0401i)T \)
3 \( 1 + (0.817 - 0.575i)T \)
5 \( 1 + (0.216 - 0.976i)T \)
7 \( 1 + (-0.900 + 0.434i)T \)
11 \( 1 + (-0.478 - 0.877i)T \)
13 \( 1 + (-0.460 + 0.887i)T \)
17 \( 1 + (-0.572 + 0.820i)T \)
19 \( 1 + (0.854 + 0.519i)T \)
23 \( 1 + (0.0361 + 0.999i)T \)
29 \( 1 + (-0.797 + 0.603i)T \)
31 \( 1 + (-0.278 - 0.960i)T \)
37 \( 1 + (-0.263 + 0.964i)T \)
41 \( 1 + (-0.319 + 0.947i)T \)
43 \( 1 + (-0.617 + 0.786i)T \)
47 \( 1 + (-0.462 + 0.886i)T \)
53 \( 1 + (-0.999 - 0.0241i)T \)
59 \( 1 + (-0.990 - 0.138i)T \)
61 \( 1 + (0.329 - 0.944i)T \)
67 \( 1 + (-0.268 + 0.963i)T \)
71 \( 1 + (0.929 - 0.368i)T \)
73 \( 1 + (0.644 + 0.764i)T \)
79 \( 1 + (-0.949 - 0.313i)T \)
83 \( 1 + (-0.327 + 0.944i)T \)
89 \( 1 + (-0.0602 - 0.998i)T \)
97 \( 1 + (-0.506 - 0.861i)T \)
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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

−19.554471511458602646120314943460, −18.706617992211689344742929105174, −17.85299492061835694928589320999, −16.84166736374043033153727554598, −15.9353590497283388749584389824, −15.46354831312161276858346111126, −14.949386979546008731643925657352, −14.11198597174688319412590256626, −13.61612049529097898842516091442, −12.95102870517524439431572561572, −12.19301861849670086254963483558, −11.03523403869796214408917699574, −10.43569260164333310258123109244, −9.95831450917004876814677811794, −9.17803192389771268428608485357, −7.78625109350251173399109382905, −7.19530060222998872525200627041, −6.73960170959167344470920989855, −5.50407514424160134587838316037, −4.87475246011608165304550454235, −3.90120208836701010356565967189, −3.19021249446070097661490744900, −2.64482644000256882364128060697, −1.97560707471098458522011243746, −0.17127531570438563530739535152, 1.32998015858324648497325742173, 1.91409743478186305508427173573, 2.96699084082761767651939651054, 3.528336960435404715412352628508, 4.437272796200771379792047153178, 5.44877364447765902058758696517, 6.10047520880989705006794214811, 6.7882569121503396054897247045, 7.80864020501625856065752963576, 8.363828834324591612723834253370, 9.41360740954288650642086747813, 9.78630078579124583801693245665, 11.23640871636427229746993942736, 11.880486260776484407807629800461, 12.731770584113806151114598669802, 13.07833033337945214195193142166, 13.666550883958808214447315653479, 14.35898220958212880266987925843, 15.2655610850632803254668493570, 15.86730083236848061082781879161, 16.54258982318165834123209476017, 17.20754218017510462870089744856, 18.46694392139306562719805771195, 19.08393999335149640581251575123, 19.78478434966665311959648348857

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