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Properties

Label 1-239-239.15-r0-0-0
Degree $1$
Conductor $239$
Sign $-0.622 - 0.782i$
Analytic cond. $1.10991$
Root an. cond. $1.10991$
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  + (0.248 − 0.968i)2-s + (0.863 − 0.503i)3-s + (−0.876 − 0.480i)4-s + (0.889 − 0.457i)5-s + (−0.273 − 0.961i)6-s + (0.773 − 0.633i)7-s + (−0.683 + 0.730i)8-s + (0.492 − 0.870i)9-s + (−0.222 − 0.974i)10-s + (−0.922 − 0.385i)11-s + (−0.999 + 0.0263i)12-s + (−0.324 + 0.946i)13-s + (−0.421 − 0.906i)14-s + (0.537 − 0.843i)15-s + (0.537 + 0.843i)16-s + (−0.421 + 0.906i)17-s + ⋯
L(s)  = 1  + (0.248 − 0.968i)2-s + (0.863 − 0.503i)3-s + (−0.876 − 0.480i)4-s + (0.889 − 0.457i)5-s + (−0.273 − 0.961i)6-s + (0.773 − 0.633i)7-s + (−0.683 + 0.730i)8-s + (0.492 − 0.870i)9-s + (−0.222 − 0.974i)10-s + (−0.922 − 0.385i)11-s + (−0.999 + 0.0263i)12-s + (−0.324 + 0.946i)13-s + (−0.421 − 0.906i)14-s + (0.537 − 0.843i)15-s + (0.537 + 0.843i)16-s + (−0.421 + 0.906i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(239\)
Sign: $-0.622 - 0.782i$
Analytic conductor: \(1.10991\)
Root analytic conductor: \(1.10991\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{239} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 239,\ (0:\ ),\ -0.622 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8260968894 - 1.712788281i\)
\(L(\frac12)\) \(\approx\) \(0.8260968894 - 1.712788281i\)
\(L(1)\) \(\approx\) \(1.135673082 - 1.129199088i\)
\(L(1)\) \(\approx\) \(1.135673082 - 1.129199088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad239 \( 1 \)
good2 \( 1 + (0.248 - 0.968i)T \)
3 \( 1 + (0.863 - 0.503i)T \)
5 \( 1 + (0.889 - 0.457i)T \)
7 \( 1 + (0.773 - 0.633i)T \)
11 \( 1 + (-0.922 - 0.385i)T \)
13 \( 1 + (-0.324 + 0.946i)T \)
17 \( 1 + (-0.421 + 0.906i)T \)
19 \( 1 + (0.912 + 0.409i)T \)
23 \( 1 + (0.739 + 0.673i)T \)
29 \( 1 + (-0.991 - 0.131i)T \)
31 \( 1 + (-0.971 + 0.235i)T \)
37 \( 1 + (-0.756 - 0.653i)T \)
41 \( 1 + (0.397 + 0.917i)T \)
43 \( 1 + (-0.643 + 0.765i)T \)
47 \( 1 + (-0.559 - 0.828i)T \)
53 \( 1 + (0.492 + 0.870i)T \)
59 \( 1 + (-0.515 + 0.857i)T \)
61 \( 1 + (-0.559 + 0.828i)T \)
67 \( 1 + (0.932 - 0.361i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.702 - 0.711i)T \)
83 \( 1 + (0.298 - 0.954i)T \)
89 \( 1 + (-0.941 - 0.336i)T \)
97 \( 1 + (0.977 - 0.209i)T \)
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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

−26.316972388622162947718785527009, −25.55521478316461124921669937856, −24.82647917171342335415424353709, −24.204616670252752645795100581857, −22.56017810862964294070280425131, −22.1229061531297717765943415071, −21.026112301708805224631107423731, −20.44248537427240161883958583320, −18.615625721454492722538138008851, −18.15190802409699907776216355119, −17.1469584144449738494877704511, −15.80520648755971097742869268764, −15.154037860438558768208138626567, −14.42087025665210958511012898549, −13.54343672352432479878067223830, −12.706881944226388571047578482254, −10.91661187106289580991245391328, −9.746121460915061714527569935651, −8.9802554590500555779174351953, −7.87968395843374476149101303334, −7.04810937042624112004816276652, −5.28896803975537958355825754848, −5.05284305977682569121366731781, −3.22840521903384912198588230894, −2.27962796783866786347485305174, 1.37353403862646890285710720221, 2.05635559267645783693576060503, 3.40909100525469268640923106756, 4.6320618223089809960313140025, 5.76119639826694448209479569237, 7.39554387090111855759000202832, 8.561137028007543531641737233591, 9.38909614737518353531302580777, 10.39399714038858674283178329310, 11.50626552278633560483688557084, 12.78295623483839817886207681592, 13.43048197135560181274327123754, 14.11596042364005291770360037760, 14.967810951687554188549668790511, 16.72902281886730880966678508831, 17.84994848497782089361376121683, 18.473218038145563943991736148936, 19.59960903726664530499971545587, 20.37520787363612978647887634749, 21.225453697863697685016559966952, 21.58935919128416243394552921406, 23.26733235750409263415908956592, 24.085813026858006580781587771407, 24.6627148914178518225105723789, 26.24281952567074532575072525512

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