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Properties

Label 1-2349-2349.1121-r0-0-0
Degree $1$
Conductor $2349$
Sign $0.991 + 0.127i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (0.981 − 0.189i)2-s + (0.927 − 0.373i)4-s + (0.991 − 0.132i)5-s + (0.140 + 0.990i)7-s + (0.840 − 0.542i)8-s + (0.947 − 0.318i)10-s + (−0.971 + 0.238i)11-s + (0.983 − 0.181i)13-s + (0.326 + 0.945i)14-s + (0.721 − 0.692i)16-s + (0.642 + 0.766i)17-s + (0.999 + 0.0249i)19-s + (0.870 − 0.492i)20-s + (−0.908 + 0.418i)22-s + (−0.933 − 0.357i)23-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (0.927 − 0.373i)4-s + (0.991 − 0.132i)5-s + (0.140 + 0.990i)7-s + (0.840 − 0.542i)8-s + (0.947 − 0.318i)10-s + (−0.971 + 0.238i)11-s + (0.983 − 0.181i)13-s + (0.326 + 0.945i)14-s + (0.721 − 0.692i)16-s + (0.642 + 0.766i)17-s + (0.999 + 0.0249i)19-s + (0.870 − 0.492i)20-s + (−0.908 + 0.418i)22-s + (−0.933 − 0.357i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.306214436 + 0.2753397773i\)
\(L(\frac12)\) \(\approx\) \(4.306214436 + 0.2753397773i\)
\(L(1)\) \(\approx\) \(2.418026642 - 0.04022057855i\)
\(L(1)\) \(\approx\) \(2.418026642 - 0.04022057855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.981 - 0.189i)T \)
5 \( 1 + (0.991 - 0.132i)T \)
7 \( 1 + (0.140 + 0.990i)T \)
11 \( 1 + (-0.971 + 0.238i)T \)
13 \( 1 + (0.983 - 0.181i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (0.999 + 0.0249i)T \)
23 \( 1 + (-0.933 - 0.357i)T \)
31 \( 1 + (-0.214 + 0.976i)T \)
37 \( 1 + (0.388 + 0.921i)T \)
41 \( 1 + (-0.802 - 0.597i)T \)
43 \( 1 + (-0.924 + 0.380i)T \)
47 \( 1 + (0.897 + 0.441i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (-0.893 - 0.448i)T \)
61 \( 1 + (-0.373 + 0.927i)T \)
67 \( 1 + (-0.107 - 0.994i)T \)
71 \( 1 + (0.542 - 0.840i)T \)
73 \( 1 + (0.715 + 0.698i)T \)
79 \( 1 + (-0.760 - 0.649i)T \)
83 \( 1 + (0.254 - 0.966i)T \)
89 \( 1 + (-0.0995 + 0.995i)T \)
97 \( 1 + (0.995 - 0.0912i)T \)
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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

−20.12386655818025384808024250005, −18.6144433065508484047572678014, −18.217812072395308480489500959582, −17.24953124247832282548911352955, −16.548110549296602239885086999749, −16.03642016375215405119168382330, −15.21436804009631179395890910243, −14.19381521436580373666892989513, −13.70402579616974432946030772366, −13.474054318104576215870358044014, −12.56558615791727689340581588433, −11.5666153447953315291470437960, −10.93074331164662681181216405436, −10.21810817335749661495317855559, −9.52818938673811353719725539007, −8.24543159469690696356819483692, −7.53806913759884786034158626819, −6.87788835047656078801428744856, −5.86433146365102961169831109872, −5.494676390297066000433562853784, −4.566360958834359158892748308250, −3.62995754915035327293979010753, −2.94036185693869440983330429491, −1.96739569226797623616348835947, −1.053803355091093768495263930345, 1.331995719221978994860641122342, 1.957776637733433700235704821446, 2.85976148055101442844822075970, 3.51247495803739210264230175203, 4.81042726620547303725966577435, 5.3572266429677402480994485555, 5.96510406718019266357698344277, 6.56582356887906793550962460470, 7.79054661926192542008263474946, 8.48074405497498372450549096106, 9.54762606510003624615047778981, 10.27758263458877867818770754157, 10.83466154782858072797852996762, 11.90905612197731230572029008104, 12.41325289453522332346030575132, 13.18056649668661647252328772431, 13.73114528905686808271925143729, 14.4697274299839724220361423552, 15.22136485163416233517243488003, 15.894707315943955029539455797000, 16.47659295615945762725749873550, 17.535447245668973992361251879460, 18.42396543060734831561331700907, 18.63075063806326362123461661018, 19.91872120083538195742018955415

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