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Properties

Label 1-1259-1259.1032-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.459 - 0.888i$
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  + (0.578 + 0.815i)2-s + (−0.822 + 0.568i)3-s + (−0.330 + 0.943i)4-s + (−0.970 − 0.242i)5-s + (−0.939 − 0.342i)6-s + (−0.586 + 0.809i)7-s + (−0.961 + 0.276i)8-s + (0.354 − 0.935i)9-s + (−0.363 − 0.931i)10-s + (0.0274 − 0.999i)11-s + (−0.264 − 0.964i)12-s + (0.839 + 0.543i)13-s + (−0.999 − 0.00998i)14-s + (0.936 − 0.351i)15-s + (−0.781 − 0.624i)16-s + (0.704 + 0.709i)17-s + ⋯
L(s)  = 1  + (0.578 + 0.815i)2-s + (−0.822 + 0.568i)3-s + (−0.330 + 0.943i)4-s + (−0.970 − 0.242i)5-s + (−0.939 − 0.342i)6-s + (−0.586 + 0.809i)7-s + (−0.961 + 0.276i)8-s + (0.354 − 0.935i)9-s + (−0.363 − 0.931i)10-s + (0.0274 − 0.999i)11-s + (−0.264 − 0.964i)12-s + (0.839 + 0.543i)13-s + (−0.999 − 0.00998i)14-s + (0.936 − 0.351i)15-s + (−0.781 − 0.624i)16-s + (0.704 + 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1032, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.459 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5312155150 + 0.8729342634i\)
\(L(\frac12)\) \(\approx\) \(-0.5312155150 + 0.8729342634i\)
\(L(1)\) \(\approx\) \(0.5454486981 + 0.6297429913i\)
\(L(1)\) \(\approx\) \(0.5454486981 + 0.6297429913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.578 + 0.815i)T \)
3 \( 1 + (-0.822 + 0.568i)T \)
5 \( 1 + (-0.970 - 0.242i)T \)
7 \( 1 + (-0.586 + 0.809i)T \)
11 \( 1 + (0.0274 - 0.999i)T \)
13 \( 1 + (0.839 + 0.543i)T \)
17 \( 1 + (0.704 + 0.709i)T \)
19 \( 1 + (-0.131 - 0.991i)T \)
23 \( 1 + (-0.0773 + 0.997i)T \)
29 \( 1 + (0.911 + 0.411i)T \)
31 \( 1 + (0.0972 + 0.995i)T \)
37 \( 1 + (0.171 + 0.985i)T \)
41 \( 1 + (-0.985 + 0.169i)T \)
43 \( 1 + (0.637 + 0.770i)T \)
47 \( 1 + (-0.991 + 0.129i)T \)
53 \( 1 + (0.112 + 0.993i)T \)
59 \( 1 + (-0.847 - 0.530i)T \)
61 \( 1 + (0.574 + 0.818i)T \)
67 \( 1 + (0.141 - 0.989i)T \)
71 \( 1 + (0.122 - 0.992i)T \)
73 \( 1 + (-0.502 + 0.864i)T \)
79 \( 1 + (0.909 - 0.416i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (0.844 + 0.534i)T \)
97 \( 1 + (0.928 + 0.370i)T \)
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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

−20.399082987479001004943167620926, −19.61983722163735518407632112181, −18.85191636911737976500373709269, −18.37988000203373154295828894838, −17.4758163567991580092058322458, −16.39074740857999962793877641598, −15.835158824244031979997152357201, −14.8152737363316154151427203262, −13.9844413067835902030031425439, −13.11299658804595029724421506009, −12.4519845411898787551402009147, −11.942120055883640228658996055087, −11.08946369510109747685948414102, −10.361668139631869592321758718352, −9.86951328101074541480743700512, −8.33464793982887050082487336852, −7.43578533559461262327061055707, −6.65692684668097881308641656182, −5.87266453315115602161938772255, −4.77454949992561855742762793925, −4.06670451942652974293082411800, −3.24881425362536222545116442513, −2.08063765973431037095043642393, −0.8573480096470437037752197151, −0.28877991948242517212800255605, 1.01189626311741606925785542544, 3.21906219515374902267871644896, 3.49088970549678040274047635462, 4.61604199899042171761054912130, 5.28539626520978253920878633109, 6.24792415849987434965339206614, 6.63379113833618488706775495647, 7.92766818027285566778809808331, 8.72935895288841428548243268806, 9.29510830429124430923461938529, 10.66664781093040076557212623452, 11.56956258233065914731538689400, 11.99667734890531583450301550688, 12.8239537036110735728554364478, 13.652224037532243179942457207472, 14.7879582538262062077269248116, 15.48627512831209354985213469575, 16.0185895469468600230637504589, 16.42548458310522857277614395359, 17.26404982595852979936986709235, 18.22783678522856250526211491268, 18.95775707280840982191925239675, 19.83803808744975146455415480551, 21.17914056213556149745159871084, 21.50975964829746886008666818000

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