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Properties

Label 1-709-709.26-r0-0-0
Degree $1$
Conductor $709$
Sign $0.413 - 0.910i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (0.560 − 0.828i)2-s + (−0.167 + 0.985i)3-s + (−0.372 − 0.928i)4-s + (0.984 − 0.176i)5-s + (0.722 + 0.691i)6-s + (0.879 − 0.476i)7-s + (−0.977 − 0.211i)8-s + (−0.943 − 0.330i)9-s + (0.405 − 0.914i)10-s + (0.823 − 0.567i)11-s + (0.977 − 0.211i)12-s + (0.0266 + 0.999i)13-s + (0.0974 − 0.995i)14-s + (0.00887 + 0.999i)15-s + (−0.722 + 0.691i)16-s + (0.903 − 0.429i)17-s + ⋯
L(s)  = 1  + (0.560 − 0.828i)2-s + (−0.167 + 0.985i)3-s + (−0.372 − 0.928i)4-s + (0.984 − 0.176i)5-s + (0.722 + 0.691i)6-s + (0.879 − 0.476i)7-s + (−0.977 − 0.211i)8-s + (−0.943 − 0.330i)9-s + (0.405 − 0.914i)10-s + (0.823 − 0.567i)11-s + (0.977 − 0.211i)12-s + (0.0266 + 0.999i)13-s + (0.0974 − 0.995i)14-s + (0.00887 + 0.999i)15-s + (−0.722 + 0.691i)16-s + (0.903 − 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.413 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897488766 - 1.222900090i\)
\(L(\frac12)\) \(\approx\) \(1.897488766 - 1.222900090i\)
\(L(1)\) \(\approx\) \(1.515607544 - 0.5624838227i\)
\(L(1)\) \(\approx\) \(1.515607544 - 0.5624838227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.560 - 0.828i)T \)
3 \( 1 + (-0.167 + 0.985i)T \)
5 \( 1 + (0.984 - 0.176i)T \)
7 \( 1 + (0.879 - 0.476i)T \)
11 \( 1 + (0.823 - 0.567i)T \)
13 \( 1 + (0.0266 + 0.999i)T \)
17 \( 1 + (0.903 - 0.429i)T \)
19 \( 1 + (-0.305 - 0.952i)T \)
23 \( 1 + (-0.781 - 0.624i)T \)
29 \( 1 + (-0.917 + 0.396i)T \)
31 \( 1 + (0.870 + 0.492i)T \)
37 \( 1 + (-0.879 + 0.476i)T \)
41 \( 1 + (0.973 + 0.228i)T \)
43 \( 1 + (0.421 - 0.906i)T \)
47 \( 1 + (-0.931 + 0.364i)T \)
53 \( 1 + (-0.861 + 0.507i)T \)
59 \( 1 + (0.977 + 0.211i)T \)
61 \( 1 + (-0.631 - 0.775i)T \)
67 \( 1 + (-0.271 - 0.962i)T \)
71 \( 1 + (0.405 + 0.914i)T \)
73 \( 1 + (-0.734 + 0.678i)T \)
79 \( 1 + (-0.254 + 0.967i)T \)
83 \( 1 + (0.998 - 0.0532i)T \)
89 \( 1 + (-0.0443 - 0.999i)T \)
97 \( 1 + (0.468 + 0.883i)T \)
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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

−22.66803579395075343862544376151, −22.370401106155641565249814354871, −21.166146174781366395744403222691, −20.677734327140127026614170853409, −19.30079902331738009020324761493, −18.33312504978282634863145842495, −17.557812335686594772362915505997, −17.40244063173152256027357370775, −16.357432193174977249034546512, −14.92066287703960540606866773038, −14.55857545455337233588708436453, −13.79736166122208569951999013002, −12.87418503107202774821853604148, −12.25554136253165200234062045351, −11.44539180605394848763794172080, −10.09965648875996820890377658990, −8.98243245530082929868195940117, −7.98413045356016346843153537203, −7.49311005286702173298443435555, −6.1555255155855317584337442215, −5.882383216711676550021752072209, −4.98120602475961425584297640117, −3.58381102948647805946529189875, −2.327278169518132635334543089895, −1.45520131355658665423368050750, 1.030205319227954556856487658811, 2.09390848717360845119878860524, 3.27161152565498360602546370770, 4.303164664212427480739712831233, 4.902837149623248485570720114057, 5.81104407478333931572627212618, 6.694210548769742898009152669213, 8.55597415724868667570424946401, 9.246121442250534347416394684742, 9.98848967245468530416922273450, 10.84454325698296430521503575571, 11.46696438131634518045182216775, 12.29686484277265664168338659519, 13.64126329240957623562663067736, 14.235192453487237372876774598863, 14.56432696833380587416278812721, 15.8862432858651265046254424212, 16.85857280785761326699347251089, 17.45587392719633349472125046806, 18.44682676782869257350769541385, 19.477859920683759892359007841802, 20.43411493728794018637716094760, 20.9668267040119842664227927658, 21.583053957097387935399982956317, 22.127878509372164850501728895841

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