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Properties

Label 1-709-709.85-r1-0-0
Degree $1$
Conductor $709$
Sign $0.543 - 0.839i$
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  + (−0.838 − 0.545i)2-s + (0.254 − 0.967i)3-s + (0.405 + 0.914i)4-s + (−0.924 − 0.380i)5-s + (−0.740 + 0.671i)6-s + (0.150 − 0.988i)7-s + (0.159 − 0.987i)8-s + (−0.870 − 0.492i)9-s + (0.567 + 0.823i)10-s + (−0.997 − 0.0709i)11-s + (0.987 − 0.159i)12-s + (−0.364 + 0.931i)13-s + (−0.665 + 0.746i)14-s + (−0.603 + 0.797i)15-s + (−0.671 + 0.740i)16-s + (−0.828 + 0.560i)17-s + ⋯
L(s)  = 1  + (−0.838 − 0.545i)2-s + (0.254 − 0.967i)3-s + (0.405 + 0.914i)4-s + (−0.924 − 0.380i)5-s + (−0.740 + 0.671i)6-s + (0.150 − 0.988i)7-s + (0.159 − 0.987i)8-s + (−0.870 − 0.492i)9-s + (0.567 + 0.823i)10-s + (−0.997 − 0.0709i)11-s + (0.987 − 0.159i)12-s + (−0.364 + 0.931i)13-s + (−0.665 + 0.746i)14-s + (−0.603 + 0.797i)15-s + (−0.671 + 0.740i)16-s + (−0.828 + 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.543 - 0.839i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.543 - 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4613980589 - 0.2509675508i\)
\(L(\frac12)\) \(\approx\) \(0.4613980589 - 0.2509675508i\)
\(L(1)\) \(\approx\) \(0.4363643337 - 0.3174730755i\)
\(L(1)\) \(\approx\) \(0.4363643337 - 0.3174730755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.838 - 0.545i)T \)
3 \( 1 + (0.254 - 0.967i)T \)
5 \( 1 + (-0.924 - 0.380i)T \)
7 \( 1 + (0.150 - 0.988i)T \)
11 \( 1 + (-0.997 - 0.0709i)T \)
13 \( 1 + (-0.364 + 0.931i)T \)
17 \( 1 + (-0.828 + 0.560i)T \)
19 \( 1 + (0.631 - 0.775i)T \)
23 \( 1 + (-0.857 + 0.515i)T \)
29 \( 1 + (0.0443 + 0.999i)T \)
31 \( 1 + (0.123 - 0.992i)T \)
37 \( 1 + (0.988 + 0.150i)T \)
41 \( 1 + (0.906 + 0.421i)T \)
43 \( 1 + (-0.322 - 0.946i)T \)
47 \( 1 + (-0.484 + 0.874i)T \)
53 \( 1 + (0.921 - 0.388i)T \)
59 \( 1 + (-0.987 - 0.159i)T \)
61 \( 1 + (-0.928 + 0.372i)T \)
67 \( 1 + (-0.943 + 0.330i)T \)
71 \( 1 + (0.567 - 0.823i)T \)
73 \( 1 + (-0.847 + 0.530i)T \)
79 \( 1 + (0.998 + 0.0620i)T \)
83 \( 1 + (0.678 - 0.734i)T \)
89 \( 1 + (0.995 + 0.0974i)T \)
97 \( 1 + (-0.476 - 0.879i)T \)
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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

−22.72815135098735497161659862856, −21.72517710887478655110268051172, −20.66715233714212498895189365981, −20.025446765681394699232531080908, −19.30667403558498065182946527557, −18.219958510031810316391976302226, −17.930918698518105286687176406662, −16.4482572120364085463831625299, −15.93544356812427577377669274892, −15.281195659827211600882455755389, −14.85705293705701492196734948284, −13.82880851964687911628266292784, −12.27945362673848546912213954744, −11.410661677540063429675572899490, −10.59772031162757870411876157443, −9.91197580858868880454051462057, −8.95231150983502339878292488009, −8.0420597781719048696802318124, −7.70447628522392761313291648657, −6.20588329328597881491988233720, −5.32344334364320674194966513256, −4.50580457941420844468081895234, −3.004489639537139012406973708379, −2.35051103371115419350898299045, −0.28754720222772780847218681443, 0.546673291938783941297104625827, 1.57258734694525920439034804781, 2.64459328370893888509831921036, 3.72112229025166115470183857571, 4.66473941414038888836116146355, 6.40613347681504237508289112656, 7.47169075516487503884765547461, 7.660749110067954435200812434155, 8.64965591910375065251545167619, 9.492504519191443499351480021543, 10.7715900237306695477984542168, 11.40078039294966945373821706289, 12.15752866303392171967850494406, 13.106188108340901259477930768003, 13.60658000641984595664314177861, 14.93486143928334561910978803946, 16.00962620712816581320659087075, 16.7131219059709717912824089625, 17.61173490472175275550056523568, 18.27986665096531541095436584925, 19.15643161339470320084517708687, 19.88606858208501956654311117013, 20.16121648734944171472247310653, 21.12816439420499782335060512077, 22.216841713998919911734337529398

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