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Properties

Label 1-1259-1259.1034-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.306 - 0.951i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (−0.653 + 0.757i)2-s + (0.590 + 0.806i)3-s + (−0.146 − 0.989i)4-s + (−0.774 − 0.632i)5-s + (−0.996 − 0.0798i)6-s + (−0.664 + 0.747i)7-s + (0.844 + 0.534i)8-s + (−0.302 + 0.953i)9-s + (0.984 − 0.173i)10-s + (0.686 + 0.727i)11-s + (0.711 − 0.702i)12-s + (−0.515 − 0.856i)13-s + (−0.131 − 0.991i)14-s + (0.0524 − 0.998i)15-s + (−0.956 + 0.290i)16-s + (0.858 + 0.513i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.757i)2-s + (0.590 + 0.806i)3-s + (−0.146 − 0.989i)4-s + (−0.774 − 0.632i)5-s + (−0.996 − 0.0798i)6-s + (−0.664 + 0.747i)7-s + (0.844 + 0.534i)8-s + (−0.302 + 0.953i)9-s + (0.984 − 0.173i)10-s + (0.686 + 0.727i)11-s + (0.711 − 0.702i)12-s + (−0.515 − 0.856i)13-s + (−0.131 − 0.991i)14-s + (0.0524 − 0.998i)15-s + (−0.956 + 0.290i)16-s + (0.858 + 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1034, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.306 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1921614062 + 0.1399594328i\)
\(L(\frac12)\) \(\approx\) \(-0.1921614062 + 0.1399594328i\)
\(L(1)\) \(\approx\) \(0.5120968686 + 0.4217185911i\)
\(L(1)\) \(\approx\) \(0.5120968686 + 0.4217185911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.653 + 0.757i)T \)
3 \( 1 + (0.590 + 0.806i)T \)
5 \( 1 + (-0.774 - 0.632i)T \)
7 \( 1 + (-0.664 + 0.747i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (-0.515 - 0.856i)T \)
17 \( 1 + (0.858 + 0.513i)T \)
19 \( 1 + (-0.978 + 0.208i)T \)
23 \( 1 + (0.151 - 0.988i)T \)
29 \( 1 + (0.127 + 0.991i)T \)
31 \( 1 + (-0.112 + 0.993i)T \)
37 \( 1 + (0.949 + 0.314i)T \)
41 \( 1 + (-0.778 - 0.628i)T \)
43 \( 1 + (0.606 + 0.794i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.889 + 0.456i)T \)
59 \( 1 + (-0.427 + 0.903i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 + (-0.0773 + 0.997i)T \)
71 \( 1 + (0.335 - 0.942i)T \)
73 \( 1 + (0.887 + 0.461i)T \)
79 \( 1 + (0.557 + 0.829i)T \)
83 \( 1 + (-0.982 + 0.183i)T \)
89 \( 1 + (-0.917 + 0.398i)T \)
97 \( 1 + (-0.311 - 0.950i)T \)
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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

−19.90281271237857197568678849568, −19.24751008791633187383502125984, −19.147999923407642998711346098973, −18.35656739739896204348315111666, −17.21471603030144965681388063495, −16.76079797463478782794413050691, −15.77000655263319460386176313719, −14.64357684499752666794485038678, −13.92841518665138121485440559039, −13.22475036081375676389932289584, −12.34924713859057230399754169039, −11.56346667152794928229613715118, −11.124959527109322929298475343016, −9.81717227570217550859104546874, −9.39322311272843590524013755835, −8.26961464875327749456153616439, −7.66103077093505148853057118022, −6.93251121980317844550216246220, −6.29772948728102185000619042798, −4.28501107036758447052960807555, −3.61352423580316663529581332541, −2.96512605504308757268807099348, −1.97568268783609856040939785600, −0.79834216293874398793315715453, −0.07277712083188634739356131851, 1.35332917549858562129118359881, 2.61942409093538647019229188991, 3.71507512767667175142945161266, 4.680304100207398748600839234106, 5.32276088286769328126082978021, 6.39360713680095410571481882037, 7.40215685790271401830676696749, 8.33827382113603392543231383213, 8.689411951778906453403392207365, 9.59465011444442555959073740478, 10.16686700021479784160413063140, 11.07086305725705016244428285378, 12.3827774255805185961208577221, 12.806839908293575581825040848130, 14.25965008544333286252112934210, 14.985208300447155257291649009675, 15.217357079892287845906202033759, 16.24029913258578922457547562048, 16.60226350306493965182527485344, 17.419385419467258729427042020596, 18.55694475351010972901542327361, 19.35538177244684615072141267414, 19.78271492371951417793728875577, 20.41360243353176056851391253394, 21.42795611686218250385339158826

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