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Properties

Label 1-709-709.71-r1-0-0
Degree $1$
Conductor $709$
Sign $-0.501 + 0.865i$
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  + (−0.962 − 0.271i)2-s + (−0.981 − 0.194i)3-s + (0.852 + 0.522i)4-s + (0.813 + 0.582i)5-s + (0.891 + 0.453i)6-s + (−0.167 + 0.985i)7-s + (−0.678 − 0.734i)8-s + (0.924 + 0.380i)9-s + (−0.624 − 0.781i)10-s + (−0.515 + 0.857i)11-s + (−0.734 − 0.678i)12-s + (−0.638 − 0.769i)13-s + (0.429 − 0.903i)14-s + (−0.684 − 0.728i)15-s + (0.453 + 0.891i)16-s + (−0.0177 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.962 − 0.271i)2-s + (−0.981 − 0.194i)3-s + (0.852 + 0.522i)4-s + (0.813 + 0.582i)5-s + (0.891 + 0.453i)6-s + (−0.167 + 0.985i)7-s + (−0.678 − 0.734i)8-s + (0.924 + 0.380i)9-s + (−0.624 − 0.781i)10-s + (−0.515 + 0.857i)11-s + (−0.734 − 0.678i)12-s + (−0.638 − 0.769i)13-s + (0.429 − 0.903i)14-s + (−0.684 − 0.728i)15-s + (0.453 + 0.891i)16-s + (−0.0177 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ -0.501 + 0.865i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3333070313 + 0.5781080705i\)
\(L(\frac12)\) \(\approx\) \(0.3333070313 + 0.5781080705i\)
\(L(1)\) \(\approx\) \(0.5550951836 + 0.09362251605i\)
\(L(1)\) \(\approx\) \(0.5550951836 + 0.09362251605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.962 - 0.271i)T \)
3 \( 1 + (-0.981 - 0.194i)T \)
5 \( 1 + (0.813 + 0.582i)T \)
7 \( 1 + (-0.167 + 0.985i)T \)
11 \( 1 + (-0.515 + 0.857i)T \)
13 \( 1 + (-0.638 - 0.769i)T \)
17 \( 1 + (-0.0177 - 0.999i)T \)
19 \( 1 + (-0.954 - 0.297i)T \)
23 \( 1 + (0.703 - 0.710i)T \)
29 \( 1 + (0.989 - 0.141i)T \)
31 \( 1 + (0.228 + 0.973i)T \)
37 \( 1 + (0.985 + 0.167i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (0.671 + 0.740i)T \)
47 \( 1 + (0.964 - 0.263i)T \)
53 \( 1 + (0.957 + 0.288i)T \)
59 \( 1 + (0.734 - 0.678i)T \)
61 \( 1 + (-0.0354 + 0.999i)T \)
67 \( 1 + (0.984 + 0.176i)T \)
71 \( 1 + (-0.624 + 0.781i)T \)
73 \( 1 + (0.507 + 0.861i)T \)
79 \( 1 + (-0.993 + 0.115i)T \)
83 \( 1 + (0.982 + 0.185i)T \)
89 \( 1 + (0.807 - 0.589i)T \)
97 \( 1 + (-0.797 + 0.603i)T \)
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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

−21.88661798150489752450779865901, −21.28979142863952964928216757844, −20.61018804713946266471078279614, −19.4327029028334096008262873689, −18.842958209413085472087633639263, −17.72840896512163863184401259935, −17.03295750333050101838083027942, −16.82117463564960264247441633698, −16.01257491101246783037095695460, −14.979918310801366095487390147305, −13.841496255620992215765775032074, −12.94100867635107333981536279120, −11.96422356527207072157756863323, −10.86819022770387700527292156397, −10.41112642644528612219359829314, −9.62667345971262570132084156415, −8.72096752723522750987761862564, −7.63337219032543098260477932602, −6.61770871116723537531242231181, −6.00015927836653810940133645492, −5.08745621874721058483344329957, −3.97733836892987024261666828466, −2.23792356513968667692651831073, −1.13464972504819451057505837779, −0.30987445507701187692568420896, 0.98526874889111250207270140581, 2.40204030035873346024404956724, 2.68818235410250763201398364371, 4.74564286925562320714416630072, 5.6464921992486633122855145858, 6.62917584873554712243171748865, 7.1612127658473827892870766449, 8.35514061162779380370991692268, 9.49928770032438958413317924464, 10.13468295607908712393303336349, 10.78355875889360538939068081707, 11.75907459625427786438321993725, 12.54929443801222480892535328643, 13.125065350139513733839697660377, 14.77813795551079855817964426719, 15.46820222069254901757340296388, 16.33305086254051360954243188524, 17.3734817717751397461537974482, 17.78686920336817791949331136027, 18.44386470004146586016424914282, 19.04874014263102491362416668944, 20.17141233568511173662469707309, 21.25244025067590510311338656341, 21.71624347273978514674268678313, 22.56047465875384519864089970602

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