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Properties

Label 1-1259-1259.1059-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.452 + 0.892i$
Analytic cond. $5.84677$
Root an. cond. $5.84677$
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  + (−0.715 − 0.699i)2-s + (0.968 + 0.247i)3-s + (0.0224 + 0.999i)4-s + (−0.656 + 0.753i)5-s + (−0.520 − 0.854i)6-s + (−0.368 + 0.929i)7-s + (0.682 − 0.730i)8-s + (0.877 + 0.478i)9-s + (0.996 − 0.0798i)10-s + (−0.405 + 0.914i)11-s + (−0.225 + 0.974i)12-s + (0.532 + 0.846i)13-s + (0.913 − 0.407i)14-s + (−0.822 + 0.568i)15-s + (−0.998 + 0.0449i)16-s + (−0.156 − 0.987i)17-s + ⋯
L(s)  = 1  + (−0.715 − 0.699i)2-s + (0.968 + 0.247i)3-s + (0.0224 + 0.999i)4-s + (−0.656 + 0.753i)5-s + (−0.520 − 0.854i)6-s + (−0.368 + 0.929i)7-s + (0.682 − 0.730i)8-s + (0.877 + 0.478i)9-s + (0.996 − 0.0798i)10-s + (−0.405 + 0.914i)11-s + (−0.225 + 0.974i)12-s + (0.532 + 0.846i)13-s + (0.913 − 0.407i)14-s + (−0.822 + 0.568i)15-s + (−0.998 + 0.0449i)16-s + (−0.156 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.452 + 0.892i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.452 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5766073622 + 0.9386005816i\)
\(L(\frac12)\) \(\approx\) \(0.5766073622 + 0.9386005816i\)
\(L(1)\) \(\approx\) \(0.8526948946 + 0.2601805701i\)
\(L(1)\) \(\approx\) \(0.8526948946 + 0.2601805701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.715 - 0.699i)T \)
3 \( 1 + (0.968 + 0.247i)T \)
5 \( 1 + (-0.656 + 0.753i)T \)
7 \( 1 + (-0.368 + 0.929i)T \)
11 \( 1 + (-0.405 + 0.914i)T \)
13 \( 1 + (0.532 + 0.846i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (-0.749 + 0.662i)T \)
23 \( 1 + (0.993 - 0.109i)T \)
29 \( 1 + (0.524 + 0.851i)T \)
31 \( 1 + (0.582 + 0.812i)T \)
37 \( 1 + (-0.578 - 0.815i)T \)
41 \( 1 + (0.660 + 0.750i)T \)
43 \( 1 + (0.711 + 0.702i)T \)
47 \( 1 + (0.675 - 0.737i)T \)
53 \( 1 + (-0.00749 + 0.999i)T \)
59 \( 1 + (-0.0673 + 0.997i)T \)
61 \( 1 + (-0.844 - 0.534i)T \)
67 \( 1 + (-0.953 - 0.299i)T \)
71 \( 1 + (-0.414 - 0.910i)T \)
73 \( 1 + (0.990 - 0.139i)T \)
79 \( 1 + (0.690 - 0.723i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (0.141 + 0.989i)T \)
97 \( 1 + (-0.972 + 0.232i)T \)
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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

−20.60649456963410757879727743527, −19.82350905759420063636767185055, −19.14487069202502381038172322835, −18.94040872353047251148359934598, −17.49956497394078340855415886088, −17.112016144941994031601688877997, −16.05878054610434812291576059352, −15.53213604341035317734522711720, −14.96207792395273159668406596236, −13.74342393564737672299230401462, −13.34946311207501588617593769429, −12.583372460452794923547031882936, −11.100140970384843622466299542880, −10.52859657488554969337182706672, −9.55679598470044111135219817569, −8.618996553367315438733534005496, −8.265006070269484737940716710787, −7.55494288714795559632466239059, −6.69006385012727920791201451761, −5.76897779365729609734394247505, −4.53053072334203682078519506485, −3.7772204662527196757452916310, −2.67341336800400320029149542442, −1.2393993195846143176040873856, −0.53797552497353013147079291259, 1.55793648494575243280933117081, 2.580673263731011281740912351506, 2.96275278486376023519486357086, 4.00078258839847671318987845625, 4.77429398521163454680871184415, 6.56447339709271755164419499468, 7.2324849722620688224543929896, 8.03126132065009471007003219355, 8.942868063069591974509746051747, 9.32736147893126558662117234754, 10.39581807341405160636864078531, 10.91366085955161439338203986539, 12.0736182329555273916626009720, 12.497096853375991655215539588528, 13.5481691026704977179582878232, 14.444017076291087160269365118998, 15.29508488283467077082068856707, 15.890795243066822174130012296043, 16.557469279000492457924919004355, 18.11588424920666744807696895782, 18.298021849329510810138257953309, 19.26276451221882857817636921165, 19.47875850877063112848042016337, 20.51144178347642720498350225452, 21.18094580283537045262442697229

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