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Properties

Label 1-709-709.21-r0-0-0
Degree $1$
Conductor $709$
Sign $0.638 - 0.769i$
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  + (−0.405 + 0.914i)2-s + (−0.870 + 0.492i)3-s + (−0.671 − 0.740i)4-s + (0.710 − 0.703i)5-s + (−0.0974 − 0.995i)6-s + (−0.954 + 0.297i)7-s + (0.949 − 0.314i)8-s + (0.515 − 0.857i)9-s + (0.355 + 0.934i)10-s + (0.989 − 0.141i)11-s + (0.949 + 0.314i)12-s + (0.734 − 0.678i)13-s + (0.115 − 0.993i)14-s + (−0.271 + 0.962i)15-s + (−0.0974 + 0.995i)16-s + (−0.372 − 0.928i)17-s + ⋯
L(s)  = 1  + (−0.405 + 0.914i)2-s + (−0.870 + 0.492i)3-s + (−0.671 − 0.740i)4-s + (0.710 − 0.703i)5-s + (−0.0974 − 0.995i)6-s + (−0.954 + 0.297i)7-s + (0.949 − 0.314i)8-s + (0.515 − 0.857i)9-s + (0.355 + 0.934i)10-s + (0.989 − 0.141i)11-s + (0.949 + 0.314i)12-s + (0.734 − 0.678i)13-s + (0.115 − 0.993i)14-s + (−0.271 + 0.962i)15-s + (−0.0974 + 0.995i)16-s + (−0.372 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.638 - 0.769i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.638 - 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5131018090 - 0.2409253358i\)
\(L(\frac12)\) \(\approx\) \(0.5131018090 - 0.2409253358i\)
\(L(1)\) \(\approx\) \(0.6095252721 + 0.1274060521i\)
\(L(1)\) \(\approx\) \(0.6095252721 + 0.1274060521i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.405 + 0.914i)T \)
3 \( 1 + (-0.870 + 0.492i)T \)
5 \( 1 + (0.710 - 0.703i)T \)
7 \( 1 + (-0.954 + 0.297i)T \)
11 \( 1 + (0.989 - 0.141i)T \)
13 \( 1 + (0.734 - 0.678i)T \)
17 \( 1 + (-0.372 - 0.928i)T \)
19 \( 1 + (-0.202 + 0.979i)T \)
23 \( 1 + (-0.468 - 0.883i)T \)
29 \( 1 + (-0.996 - 0.0886i)T \)
31 \( 1 + (0.969 - 0.245i)T \)
37 \( 1 + (-0.954 + 0.297i)T \)
41 \( 1 + (-0.645 + 0.764i)T \)
43 \( 1 + (-0.792 - 0.610i)T \)
47 \( 1 + (-0.530 + 0.847i)T \)
53 \( 1 + (-0.697 - 0.716i)T \)
59 \( 1 + (0.949 - 0.314i)T \)
61 \( 1 + (-0.722 - 0.691i)T \)
67 \( 1 + (0.781 + 0.624i)T \)
71 \( 1 + (0.355 - 0.934i)T \)
73 \( 1 + (-0.437 - 0.899i)T \)
79 \( 1 + (-0.992 + 0.123i)T \)
83 \( 1 + (0.0797 - 0.996i)T \)
89 \( 1 + (-0.981 + 0.194i)T \)
97 \( 1 + (0.545 + 0.838i)T \)
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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

−22.59694871485267680370181297259, −21.82986370528621561511638452313, −21.446278367326561764433104593665, −20.040656975125575100767039606212, −19.2765639731878445975404527707, −18.802495389874775712727790404376, −17.82090478218247231062198196266, −17.25212600505008837733772327080, −16.653021324090112361479511488791, −15.493634186787712645834912681127, −13.97702695221180348860145958295, −13.44450509448740443549163799418, −12.728255185210625552952328045424, −11.69295575933142139994853646751, −11.085356068602192377455238532, −10.25436540407845462494962220245, −9.54614828992142464651333499402, −8.5984561620555912498348179127, −7.08491217133443010863137113681, −6.6495360504466919489454738973, −5.66721564797425476831045917807, −4.241891090001854184958598040987, −3.39177802001318098195446747063, −2.06570446242446604101457410242, −1.32857462511161687985698133529, 0.38277758203617566381421891610, 1.5704188681950923868955028147, 3.54017130137541922986872136146, 4.59556475458779395382421068299, 5.5095800945948009049296375356, 6.24842590062141819945280975692, 6.67007758106506237654846730324, 8.2621084191039777272339962802, 9.09585538606702317300563505970, 9.78261166719223936484475703324, 10.373624325750599867615179168164, 11.66144681440552002413996188162, 12.656702552627644494775133627521, 13.41739263581065653757788973719, 14.40663116194250377571045604097, 15.4792771281963389680797364496, 16.152645616186510592913207996488, 16.70279683615431843650815236335, 17.344300884508149176590491013371, 18.22039024616868915107229232963, 18.88196417555403269893715814570, 20.13905511219077950448371630004, 20.86138178087130730630388806918, 22.17755474632313126643113465439, 22.46138716635727126511825774552

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