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Properties

Label 1-1259-1259.1031-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.802 - 0.596i$
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  + (0.606 + 0.794i)2-s + (−0.340 + 0.940i)3-s + (−0.264 + 0.964i)4-s + (0.745 + 0.666i)5-s + (−0.953 + 0.299i)6-s + (0.992 − 0.119i)7-s + (−0.926 + 0.375i)8-s + (−0.768 − 0.639i)9-s + (−0.0773 + 0.997i)10-s + (−0.586 + 0.809i)11-s + (−0.817 − 0.576i)12-s + (−0.386 + 0.922i)13-s + (0.697 + 0.716i)14-s + (−0.880 + 0.474i)15-s + (−0.860 − 0.509i)16-s + (0.955 + 0.295i)17-s + ⋯
L(s)  = 1  + (0.606 + 0.794i)2-s + (−0.340 + 0.940i)3-s + (−0.264 + 0.964i)4-s + (0.745 + 0.666i)5-s + (−0.953 + 0.299i)6-s + (0.992 − 0.119i)7-s + (−0.926 + 0.375i)8-s + (−0.768 − 0.639i)9-s + (−0.0773 + 0.997i)10-s + (−0.586 + 0.809i)11-s + (−0.817 − 0.576i)12-s + (−0.386 + 0.922i)13-s + (0.697 + 0.716i)14-s + (−0.880 + 0.474i)15-s + (−0.860 − 0.509i)16-s + (0.955 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.802 - 0.596i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6313308653 + 1.909413607i\)
\(L(\frac12)\) \(\approx\) \(-0.6313308653 + 1.909413607i\)
\(L(1)\) \(\approx\) \(0.6675500444 + 1.266539540i\)
\(L(1)\) \(\approx\) \(0.6675500444 + 1.266539540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.606 + 0.794i)T \)
3 \( 1 + (-0.340 + 0.940i)T \)
5 \( 1 + (0.745 + 0.666i)T \)
7 \( 1 + (0.992 - 0.119i)T \)
11 \( 1 + (-0.586 + 0.809i)T \)
13 \( 1 + (-0.386 + 0.922i)T \)
17 \( 1 + (0.955 + 0.295i)T \)
19 \( 1 + (-0.395 + 0.918i)T \)
23 \( 1 + (0.995 - 0.0897i)T \)
29 \( 1 + (-0.828 + 0.559i)T \)
31 \( 1 + (0.0623 - 0.998i)T \)
37 \( 1 + (0.344 - 0.938i)T \)
41 \( 1 + (0.524 - 0.851i)T \)
43 \( 1 + (0.354 + 0.935i)T \)
47 \( 1 + (-0.570 + 0.821i)T \)
53 \( 1 + (-0.907 - 0.420i)T \)
59 \( 1 + (0.718 + 0.695i)T \)
61 \( 1 + (0.249 + 0.968i)T \)
67 \( 1 + (0.382 - 0.924i)T \)
71 \( 1 + (-0.934 - 0.356i)T \)
73 \( 1 + (-0.254 - 0.967i)T \)
79 \( 1 + (-0.981 - 0.193i)T \)
83 \( 1 + (-0.273 - 0.961i)T \)
89 \( 1 + (0.391 + 0.920i)T \)
97 \( 1 + (0.498 - 0.866i)T \)
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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

−20.5956902726155672289674369811, −20.05118259981785823680963224818, −18.96204983721922608623671904663, −18.51782077773306297965558906626, −17.55554453884613350294867827666, −17.19914141797728948316214549722, −15.98698270409852690953301961583, −14.85332668193595696525105169493, −14.17601453746481481840318793417, −13.31933864754140172422213695975, −12.98055297041753004218879102549, −12.14763072225825956070145160832, −11.33175510185511292233086676974, −10.73991265505136749756332543819, −9.78161349234539820332703011230, −8.680550436998832176428033344638, −8.05554487458681339843363264801, −6.88191385438747087902319351500, −5.702397778742441598423873679197, −5.3521534706700249978346549685, −4.701607107301494144830333096950, −3.06173420354804278262391106220, −2.398241000654430286391660602977, −1.37191324922585975675178805226, −0.687197340165939092998180353700, 1.85021340657015026647677062, 2.8807822524727411642160961165, 3.97019726974233693313340578692, 4.67500351282420558100815742754, 5.476269420136553420825471263999, 6.067399968124741402539004919779, 7.179463495000005864258145523319, 7.82201605013834765301260086850, 9.01367424838050502183838669612, 9.700196905307557438398178786431, 10.63897129715373698953686123625, 11.34777384078747471714579451312, 12.26838026254438470701494699951, 13.14320942042039589260602397720, 14.38538724094106191976782721939, 14.56252168169763987057378639180, 15.0727038185489110386661555562, 16.2240262316224426871014599503, 16.870337065773532470340307988244, 17.471150432651046692992093904587, 18.10478746551393145533895636109, 19.014919236570208543602383716938, 20.680461400345410908487927640153, 21.00882567490773756599811493744, 21.44268547553634844109982740511

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