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Properties

Label 1-1259-1259.1078-r0-0-0
Degree $1$
Conductor $1259$
Sign $0.973 - 0.229i$
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  + (−0.349 + 0.936i)2-s + (−0.264 + 0.964i)3-s + (−0.755 − 0.655i)4-s + (−0.423 + 0.906i)5-s + (−0.811 − 0.584i)6-s + (−0.993 + 0.114i)7-s + (0.877 − 0.478i)8-s + (−0.860 − 0.509i)9-s + (−0.700 − 0.713i)10-s + (0.489 + 0.871i)11-s + (0.831 − 0.555i)12-s + (−0.545 + 0.838i)13-s + (0.239 − 0.970i)14-s + (−0.762 − 0.647i)15-s + (0.141 + 0.989i)16-s + (−0.283 + 0.959i)17-s + ⋯
L(s)  = 1  + (−0.349 + 0.936i)2-s + (−0.264 + 0.964i)3-s + (−0.755 − 0.655i)4-s + (−0.423 + 0.906i)5-s + (−0.811 − 0.584i)6-s + (−0.993 + 0.114i)7-s + (0.877 − 0.478i)8-s + (−0.860 − 0.509i)9-s + (−0.700 − 0.713i)10-s + (0.489 + 0.871i)11-s + (0.831 − 0.555i)12-s + (−0.545 + 0.838i)13-s + (0.239 − 0.970i)14-s + (−0.762 − 0.647i)15-s + (0.141 + 0.989i)16-s + (−0.283 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2094211281 + 0.02430806962i\)
\(L(\frac12)\) \(\approx\) \(-0.2094211281 + 0.02430806962i\)
\(L(1)\) \(\approx\) \(0.2201858474 + 0.4446274783i\)
\(L(1)\) \(\approx\) \(0.2201858474 + 0.4446274783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.349 + 0.936i)T \)
3 \( 1 + (-0.264 + 0.964i)T \)
5 \( 1 + (-0.423 + 0.906i)T \)
7 \( 1 + (-0.993 + 0.114i)T \)
11 \( 1 + (0.489 + 0.871i)T \)
13 \( 1 + (-0.545 + 0.838i)T \)
17 \( 1 + (-0.283 + 0.959i)T \)
19 \( 1 + (-0.947 - 0.318i)T \)
23 \( 1 + (0.765 + 0.643i)T \)
29 \( 1 + (-0.996 - 0.0848i)T \)
31 \( 1 + (-0.377 + 0.926i)T \)
37 \( 1 + (-0.799 + 0.600i)T \)
41 \( 1 + (-0.828 - 0.559i)T \)
43 \( 1 + (-0.781 + 0.624i)T \)
47 \( 1 + (-0.00749 - 0.999i)T \)
53 \( 1 + (0.690 + 0.723i)T \)
59 \( 1 + (0.541 + 0.840i)T \)
61 \( 1 + (-0.302 - 0.953i)T \)
67 \( 1 + (0.0823 + 0.996i)T \)
71 \( 1 + (-0.537 - 0.843i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (0.836 + 0.547i)T \)
83 \( 1 + (0.445 + 0.895i)T \)
89 \( 1 + (-0.944 - 0.328i)T \)
97 \( 1 + (0.975 - 0.217i)T \)
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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

−20.20070218396473909079318164276, −19.429318483956723477068746946784, −19.040905576603122467696706680143, −18.30278629325009779474510018384, −17.16799119428127627094168560932, −16.83631789029954747114631390299, −16.13333620449877614205821098052, −14.77819075909910293942372999631, −13.62291239667309844786922536189, −13.0385851274793534249229567131, −12.61938392075516518601448904169, −11.808809669549477182211951203833, −11.163629531660399647207504943586, −10.20284898634352896638664248005, −9.131557073299366028417921027708, −8.63783823907625428454573766768, −7.73766091559487180579763089977, −6.930973877815457478332517730950, −5.78024352327681229810596221234, −4.919371569668452231168980707199, −3.75304835065735191160909856391, −2.95218429737183049673062759274, −1.93416706269728436728273057660, −0.72736320715141847250473298547, −0.1461179570741383889400909366, 1.9598666605890283625409616733, 3.40220731906548043586023985887, 4.059408479587451304823382521361, 4.917378309985570215018463635746, 6.00357485359764175466353664821, 6.77525062133070497050575024865, 7.16666240001259152516639151399, 8.59882591730739013447939760676, 9.19456208367281441312019744240, 10.04025660436765582303387785441, 10.510195998504793392257681468772, 11.51911589079465308721760618610, 12.48707918236406536873535857297, 13.56753624599806661642521457758, 14.571208505288400302921139084566, 15.206768434039062157634098380798, 15.378700039456681467309472725979, 16.553016231871762182914638257961, 16.980094119812586395690017542435, 17.73378444562598856179786232854, 18.75322338223485149390415268580, 19.48363468515876763549904125992, 19.87867062484825660915962996820, 21.35716651338787213239440091277, 22.1326358484384979852299924768

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