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Properties

Label 1-1259-1259.1062-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.763 - 0.645i$
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  + (0.0723 + 0.997i)2-s + (−0.671 + 0.740i)3-s + (−0.989 + 0.144i)4-s + (0.711 + 0.702i)5-s + (−0.787 − 0.616i)6-s + (−0.511 − 0.859i)7-s + (−0.215 − 0.976i)8-s + (−0.0972 − 0.995i)9-s + (−0.649 + 0.760i)10-s + (−0.994 + 0.104i)11-s + (0.557 − 0.829i)12-s + (0.887 − 0.461i)13-s + (0.820 − 0.572i)14-s + (−0.998 + 0.0549i)15-s + (0.958 − 0.285i)16-s + (−0.528 + 0.848i)17-s + ⋯
L(s)  = 1  + (0.0723 + 0.997i)2-s + (−0.671 + 0.740i)3-s + (−0.989 + 0.144i)4-s + (0.711 + 0.702i)5-s + (−0.787 − 0.616i)6-s + (−0.511 − 0.859i)7-s + (−0.215 − 0.976i)8-s + (−0.0972 − 0.995i)9-s + (−0.649 + 0.760i)10-s + (−0.994 + 0.104i)11-s + (0.557 − 0.829i)12-s + (0.887 − 0.461i)13-s + (0.820 − 0.572i)14-s + (−0.998 + 0.0549i)15-s + (0.958 − 0.285i)16-s + (−0.528 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1062, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2322497044 + 0.6345037870i\)
\(L(\frac12)\) \(\approx\) \(-0.2322497044 + 0.6345037870i\)
\(L(1)\) \(\approx\) \(0.4949773153 + 0.5654111559i\)
\(L(1)\) \(\approx\) \(0.4949773153 + 0.5654111559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.0723 + 0.997i)T \)
3 \( 1 + (-0.671 + 0.740i)T \)
5 \( 1 + (0.711 + 0.702i)T \)
7 \( 1 + (-0.511 - 0.859i)T \)
11 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.887 - 0.461i)T \)
17 \( 1 + (-0.528 + 0.848i)T \)
19 \( 1 + (0.975 + 0.217i)T \)
23 \( 1 + (0.999 - 0.00998i)T \)
29 \( 1 + (-0.721 + 0.691i)T \)
31 \( 1 + (0.335 + 0.942i)T \)
37 \( 1 + (-0.884 + 0.465i)T \)
41 \( 1 + (-0.594 - 0.803i)T \)
43 \( 1 + (-0.977 + 0.213i)T \)
47 \( 1 + (-0.0673 + 0.997i)T \)
53 \( 1 + (0.541 - 0.840i)T \)
59 \( 1 + (-0.907 - 0.420i)T \)
61 \( 1 + (-0.368 - 0.929i)T \)
67 \( 1 + (0.675 - 0.737i)T \)
71 \( 1 + (0.925 + 0.379i)T \)
73 \( 1 + (-0.664 - 0.747i)T \)
79 \( 1 + (0.481 + 0.876i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (0.991 + 0.129i)T \)
97 \( 1 + (-0.395 + 0.918i)T \)
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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

−20.65154183987481890977858877561, −19.8974613647694127225410521888, −18.7371997000048208024174912390, −18.52604975595204816950456442498, −17.86470453681252862522351894720, −16.951498664958258528828445032223, −16.19064543059787013056470716821, −15.27531461033948433000495284848, −13.735785570621862650421838159274, −13.402713271109088461754523228841, −12.925167988912894475248422339912, −11.92989622254138300588170787827, −11.50185954605160096066004651420, −10.539083841533184307134103739762, −9.61974439501712599621034671454, −8.93334541248938589087146098781, −8.15997109017278896207400574508, −6.90364961831767918967766017815, −5.752327833423055912257708259395, −5.405705711314661807271296660005, −4.53949277890019382543450797528, −3.035711130418457038942530024516, −2.29225779336006562835360396101, −1.43506835487590562793589803049, −0.31813102958541303942258984378, 1.22813876984271630834409032813, 3.24438411337589023597374377654, 3.60866436841799330406580395143, 4.97102734719126032623571615908, 5.44789595387778423067299201153, 6.46780415041092222127453345343, 6.852055446654538395523533325355, 7.94526208250259551122475863373, 9.02321621451041424763577373422, 9.83599411832180996625524615675, 10.54167942368084697733522309355, 10.94763563956458098838627017514, 12.50767179566457230765616864982, 13.25497431039833159388954470243, 13.858570897747413350379041231934, 14.813639250441119308655890064692, 15.55441768349551318616351923103, 16.061599459563844201277320349169, 16.98597247955557741805187128659, 17.48473013435549022951852209774, 18.21244669141766958020179863553, 18.8300726817540838530660884147, 20.19312866907443120427442047257, 21.03544961742732966935193441079, 21.69954745653757248724573025695

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