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Properties

Label 1-709-709.75-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.998 + 0.0478i$
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  + (0.574 + 0.818i)2-s + (0.0797 + 0.996i)3-s + (−0.339 + 0.940i)4-s + (0.734 + 0.678i)5-s + (−0.769 + 0.638i)6-s + (0.910 − 0.413i)7-s + (−0.964 + 0.263i)8-s + (−0.987 + 0.159i)9-s + (−0.132 + 0.991i)10-s + (0.288 + 0.957i)11-s + (−0.964 − 0.263i)12-s + (0.910 + 0.413i)13-s + (0.861 + 0.507i)14-s + (−0.617 + 0.786i)15-s + (−0.769 − 0.638i)16-s + (0.288 − 0.957i)17-s + ⋯
L(s)  = 1  + (0.574 + 0.818i)2-s + (0.0797 + 0.996i)3-s + (−0.339 + 0.940i)4-s + (0.734 + 0.678i)5-s + (−0.769 + 0.638i)6-s + (0.910 − 0.413i)7-s + (−0.964 + 0.263i)8-s + (−0.987 + 0.159i)9-s + (−0.132 + 0.991i)10-s + (0.288 + 0.957i)11-s + (−0.964 − 0.263i)12-s + (0.910 + 0.413i)13-s + (0.861 + 0.507i)14-s + (−0.617 + 0.786i)15-s + (−0.769 − 0.638i)16-s + (0.288 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.998 + 0.0478i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.998 + 0.0478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05565836910 + 2.323986415i\)
\(L(\frac12)\) \(\approx\) \(0.05565836910 + 2.323986415i\)
\(L(1)\) \(\approx\) \(0.9153601486 + 1.393776662i\)
\(L(1)\) \(\approx\) \(0.9153601486 + 1.393776662i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.574 + 0.818i)T \)
3 \( 1 + (0.0797 + 0.996i)T \)
5 \( 1 + (0.734 + 0.678i)T \)
7 \( 1 + (0.910 - 0.413i)T \)
11 \( 1 + (0.288 + 0.957i)T \)
13 \( 1 + (0.910 + 0.413i)T \)
17 \( 1 + (0.288 - 0.957i)T \)
19 \( 1 + (-0.964 + 0.263i)T \)
23 \( 1 + (0.949 - 0.314i)T \)
29 \( 1 + (-0.697 + 0.716i)T \)
31 \( 1 + (-0.617 + 0.786i)T \)
37 \( 1 + (0.910 - 0.413i)T \)
41 \( 1 + (-0.0266 - 0.999i)T \)
43 \( 1 + (-0.931 - 0.364i)T \)
47 \( 1 + (0.949 - 0.314i)T \)
53 \( 1 + (-0.617 + 0.786i)T \)
59 \( 1 + (-0.964 + 0.263i)T \)
61 \( 1 + (-0.833 + 0.552i)T \)
67 \( 1 + (0.977 + 0.211i)T \)
71 \( 1 + (-0.132 - 0.991i)T \)
73 \( 1 + (0.802 - 0.596i)T \)
79 \( 1 + (-0.437 - 0.899i)T \)
83 \( 1 + (0.658 + 0.752i)T \)
89 \( 1 + (0.185 - 0.982i)T \)
97 \( 1 + (-0.887 - 0.461i)T \)
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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

−21.91680431547549166164160517613, −21.39340953211518671072294121204, −20.64450883534965376485350245408, −19.90841886362303181324961994898, −18.87159298580779744939734390271, −18.46158824037029229470523079109, −17.44411692202033906589929812345, −16.82725252229298535912079844945, −15.22509328919454141784031010365, −14.53482879254826551756389891314, −13.60238852880874181169506469307, −13.09625577790844163108490054486, −12.45715431620768830834554099203, −11.27088906344193694398667885069, −11.01817568794568659803121853468, −9.514668320693140798593467586983, −8.62677363824436758564920058385, −8.09523711585771569048757169677, −6.26649788721125314517520981603, −5.908366864177143301547581061064, −4.98377779911738660728306408422, −3.73472166120714600635890479097, −2.551396533020658974170156006526, −1.63900794225243955234742979062, −0.975798689127859881998693731396, 1.923700031146992742101070558733, 3.14118828732622752565156827416, 4.09878575770804800155577667534, 4.88123365706409870740667728816, 5.688939656657253464330815398, 6.749330574064521647519938623143, 7.51692361599126176025142604350, 8.77670647326357453449299395537, 9.339049018655051247698499311493, 10.58688401627581265873559650035, 11.15166679726441954303939876412, 12.2981833321237561902251849502, 13.54061903971825567571360138505, 14.18360876762041955908031103094, 14.794619053961677452003022563887, 15.348958155114771641470539038803, 16.556172386416885511214910346699, 17.02567694191459111662847615476, 17.8929152899683784828036596627, 18.58549537485441639415419042341, 20.27625149564136975368390747189, 20.84877702871363234422904365224, 21.47397876717905755911764688340, 22.19420708766706070957142231093, 23.133330783409394683131171635070

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