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Properties

Label 1-1259-1259.1013-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.588 - 0.808i$
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  + (0.977 + 0.213i)2-s + (−0.972 + 0.232i)3-s + (0.909 + 0.416i)4-s + (−0.817 − 0.576i)5-s + (−0.999 + 0.0199i)6-s + (−0.839 + 0.543i)7-s + (0.799 + 0.600i)8-s + (0.891 − 0.452i)9-s + (−0.675 − 0.737i)10-s + (−0.549 + 0.835i)11-s + (−0.981 − 0.193i)12-s + (0.254 + 0.967i)13-s + (−0.936 + 0.351i)14-s + (0.928 + 0.370i)15-s + (0.653 + 0.757i)16-s + (−0.990 + 0.134i)17-s + ⋯
L(s)  = 1  + (0.977 + 0.213i)2-s + (−0.972 + 0.232i)3-s + (0.909 + 0.416i)4-s + (−0.817 − 0.576i)5-s + (−0.999 + 0.0199i)6-s + (−0.839 + 0.543i)7-s + (0.799 + 0.600i)8-s + (0.891 − 0.452i)9-s + (−0.675 − 0.737i)10-s + (−0.549 + 0.835i)11-s + (−0.981 − 0.193i)12-s + (0.254 + 0.967i)13-s + (−0.936 + 0.351i)14-s + (0.928 + 0.370i)15-s + (0.653 + 0.757i)16-s + (−0.990 + 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.588 - 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3663298044 + 0.7201604149i\)
\(L(\frac12)\) \(\approx\) \(-0.3663298044 + 0.7201604149i\)
\(L(1)\) \(\approx\) \(0.8820062450 + 0.4672546011i\)
\(L(1)\) \(\approx\) \(0.8820062450 + 0.4672546011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.977 + 0.213i)T \)
3 \( 1 + (-0.972 + 0.232i)T \)
5 \( 1 + (-0.817 - 0.576i)T \)
7 \( 1 + (-0.839 + 0.543i)T \)
11 \( 1 + (-0.549 + 0.835i)T \)
13 \( 1 + (0.254 + 0.967i)T \)
17 \( 1 + (-0.990 + 0.134i)T \)
19 \( 1 + (-0.0524 + 0.998i)T \)
23 \( 1 + (-0.937 - 0.347i)T \)
29 \( 1 + (0.911 + 0.411i)T \)
31 \( 1 + (0.934 + 0.356i)T \)
37 \( 1 + (0.996 - 0.0798i)T \)
41 \( 1 + (-0.985 + 0.169i)T \)
43 \( 1 + (0.973 - 0.227i)T \)
47 \( 1 + (0.0374 + 0.999i)T \)
53 \( 1 + (-0.618 - 0.785i)T \)
59 \( 1 + (-0.278 + 0.960i)T \)
61 \( 1 + (-0.999 - 0.0349i)T \)
67 \( 1 + (0.400 + 0.916i)T \)
71 \( 1 + (-0.302 + 0.953i)T \)
73 \( 1 + (0.992 - 0.119i)T \)
79 \( 1 + (-0.970 + 0.242i)T \)
83 \( 1 + (0.739 - 0.673i)T \)
89 \( 1 + (-0.102 + 0.994i)T \)
97 \( 1 + (0.454 - 0.890i)T \)
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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

−20.24296438093244217579717561863, −19.725935823280899288799371457751, −18.988295908247658150194768516721, −18.19467123265657572802100832537, −17.224426625922723206612266432903, −16.22598993120183069648359439536, −15.6297822779297466905421162916, −15.358604032833883306505104916807, −13.76372582708434866973680053730, −13.46036081748124471625617652336, −12.59089276840871681326403373626, −11.83647658830657289279814327026, −10.99572875232940536126113757446, −10.67869420467664111950119180554, −9.81690976236525705571142241180, −8.0976759055401283996000906619, −7.37003416507654775874565398, −6.4274602207833624214815404091, −6.11545229319320494165476535957, −4.91364615268359449265606908432, −4.16456382881183161412949846381, −3.23606996380340007226318468211, −2.47383405012693068266135084423, −0.78348446302013496546070576816, −0.168648301289282541225759866597, 1.44800572669594762075201976294, 2.62149138774338016597592727408, 3.9160928123977594284076908731, 4.38966984169340221862430905372, 5.156429953149988346237586342329, 6.18910173174181488763111097184, 6.66104955758526201599302741882, 7.65091612217071738425496562456, 8.647613002062147717468176081818, 9.78247264977093578090873801261, 10.647579629218760021307769863245, 11.598353197811109863798806686429, 12.19754216103409212836434858782, 12.60794189385740145506969448358, 13.407442199883606216168619132332, 14.58822222927916397484881429713, 15.54775000541517402580023231411, 15.90855207948366346243118042771, 16.40997537250362703518433981002, 17.271909626705033918450186030950, 18.2662715352067364379483149731, 19.181993671839828791369819697927, 20.03918604147726165944009500651, 20.80622386921126805466282836162, 21.53693920242515545466120681459

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