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Properties

Label 1-709-709.90-r1-0-0
Degree $1$
Conductor $709$
Sign $0.438 + 0.898i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (0.934 − 0.355i)2-s + (0.969 + 0.245i)3-s + (0.746 − 0.665i)4-s + (−0.603 + 0.797i)5-s + (0.993 − 0.115i)6-s + (0.997 − 0.0709i)7-s + (0.461 − 0.887i)8-s + (0.879 + 0.476i)9-s + (−0.280 + 0.959i)10-s + (0.305 + 0.952i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (0.906 − 0.421i)14-s + (−0.781 + 0.624i)15-s + (0.115 − 0.993i)16-s + (−0.740 + 0.671i)17-s + ⋯
L(s)  = 1  + (0.934 − 0.355i)2-s + (0.969 + 0.245i)3-s + (0.746 − 0.665i)4-s + (−0.603 + 0.797i)5-s + (0.993 − 0.115i)6-s + (0.997 − 0.0709i)7-s + (0.461 − 0.887i)8-s + (0.879 + 0.476i)9-s + (−0.280 + 0.959i)10-s + (0.305 + 0.952i)11-s + (0.887 − 0.461i)12-s + (−0.899 + 0.437i)13-s + (0.906 − 0.421i)14-s + (−0.781 + 0.624i)15-s + (0.115 − 0.993i)16-s + (−0.740 + 0.671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.438 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.535751794 + 2.833533731i\)
\(L(\frac12)\) \(\approx\) \(4.535751794 + 2.833533731i\)
\(L(1)\) \(\approx\) \(2.490154279 + 0.3983309515i\)
\(L(1)\) \(\approx\) \(2.490154279 + 0.3983309515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.934 - 0.355i)T \)
3 \( 1 + (0.969 + 0.245i)T \)
5 \( 1 + (-0.603 + 0.797i)T \)
7 \( 1 + (0.997 - 0.0709i)T \)
11 \( 1 + (0.305 + 0.952i)T \)
13 \( 1 + (-0.899 + 0.437i)T \)
17 \( 1 + (-0.740 + 0.671i)T \)
19 \( 1 + (0.0443 + 0.999i)T \)
23 \( 1 + (-0.838 + 0.545i)T \)
29 \( 1 + (0.924 + 0.380i)T \)
31 \( 1 + (-0.988 + 0.150i)T \)
37 \( 1 + (0.0709 + 0.997i)T \)
41 \( 1 + (0.967 + 0.254i)T \)
43 \( 1 + (0.0620 - 0.998i)T \)
47 \( 1 + (0.998 - 0.0532i)T \)
53 \( 1 + (-0.364 + 0.931i)T \)
59 \( 1 + (-0.887 - 0.461i)T \)
61 \( 1 + (-0.995 + 0.0974i)T \)
67 \( 1 + (-0.468 - 0.883i)T \)
71 \( 1 + (-0.280 - 0.959i)T \)
73 \( 1 + (-0.106 - 0.994i)T \)
79 \( 1 + (0.651 - 0.758i)T \)
83 \( 1 + (0.786 + 0.617i)T \)
89 \( 1 + (0.228 + 0.973i)T \)
97 \( 1 + (0.567 - 0.823i)T \)
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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

−22.10023076922255194539975819267, −21.45925809927067024363279637461, −20.64399563688713491339966480281, −19.94659430740776782037690338050, −19.49086666224893755619934634451, −18.086631929098316166809295337206, −17.27282826586155376289537859317, −16.14118963466946456328148767025, −15.61016277024107315686066151878, −14.67667537081268905417981684344, −14.09853817510487919947212881053, −13.27361262656929318063871817625, −12.466072530738771978262634113748, −11.70223118008682305061890706339, −10.86988445853123962966393927760, −9.21474192593526025991165272443, −8.49049052336909097282035627320, −7.76540377951950819928465940328, −7.09438268755628693930223298658, −5.75966904027124527630089846357, −4.66041264903179971753890449115, −4.18514007633732436538512662129, −2.95165782656660292937618740313, −2.104452199863818210156627689135, −0.7071789379264768454548913253, 1.6866932591155753951328090032, 2.222754283209786486158866943156, 3.40004009729960527280068347290, 4.25114385432396929740446371618, 4.76424917160278141344477630946, 6.28445679894988138757060301858, 7.37359919410377546206206168511, 7.791189580374892373354210098746, 9.176169574773636005923104138938, 10.25839984151205222412178646731, 10.78374435359907443490990977825, 11.9681112287085423737559171812, 12.429798332586427023471473870123, 13.884043156729136087749963389423, 14.23662581961172349928815067353, 15.11717441059056921975085028447, 15.30997408049523010999812765449, 16.56841430150207215182907655843, 17.88021775494725944099759005687, 18.78026425569590968738516036865, 19.70272312453144618637982922156, 20.06969219199509281142342386191, 20.933745193344310158931143482586, 21.88779449612160928397324636028, 22.20369734227073767834457800816

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