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Properties

Label 1-709-709.343-r0-0-0
Degree $1$
Conductor $709$
Sign $0.309 - 0.950i$
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  + (0.185 + 0.982i)2-s + (0.802 − 0.596i)3-s + (−0.931 + 0.364i)4-s + (0.949 − 0.314i)5-s + (0.734 + 0.678i)6-s + (−0.964 + 0.263i)7-s + (−0.530 − 0.847i)8-s + (0.288 − 0.957i)9-s + (0.484 + 0.874i)10-s + (−0.697 − 0.716i)11-s + (−0.530 + 0.847i)12-s + (−0.964 − 0.263i)13-s + (−0.437 − 0.899i)14-s + (0.574 − 0.818i)15-s + (0.734 − 0.678i)16-s + (−0.697 + 0.716i)17-s + ⋯
L(s)  = 1  + (0.185 + 0.982i)2-s + (0.802 − 0.596i)3-s + (−0.931 + 0.364i)4-s + (0.949 − 0.314i)5-s + (0.734 + 0.678i)6-s + (−0.964 + 0.263i)7-s + (−0.530 − 0.847i)8-s + (0.288 − 0.957i)9-s + (0.484 + 0.874i)10-s + (−0.697 − 0.716i)11-s + (−0.530 + 0.847i)12-s + (−0.964 − 0.263i)13-s + (−0.437 − 0.899i)14-s + (0.574 − 0.818i)15-s + (0.734 − 0.678i)16-s + (−0.697 + 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.309 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9581882529 - 0.6954923743i\)
\(L(\frac12)\) \(\approx\) \(0.9581882529 - 0.6954923743i\)
\(L(1)\) \(\approx\) \(1.135488921 + 0.03462633493i\)
\(L(1)\) \(\approx\) \(1.135488921 + 0.03462633493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.185 + 0.982i)T \)
3 \( 1 + (0.802 - 0.596i)T \)
5 \( 1 + (0.949 - 0.314i)T \)
7 \( 1 + (-0.964 + 0.263i)T \)
11 \( 1 + (-0.697 - 0.716i)T \)
13 \( 1 + (-0.964 - 0.263i)T \)
17 \( 1 + (-0.697 + 0.716i)T \)
19 \( 1 + (-0.530 - 0.847i)T \)
23 \( 1 + (-0.833 - 0.552i)T \)
29 \( 1 + (0.994 - 0.106i)T \)
31 \( 1 + (0.574 - 0.818i)T \)
37 \( 1 + (-0.964 + 0.263i)T \)
41 \( 1 + (0.977 - 0.211i)T \)
43 \( 1 + (-0.987 + 0.159i)T \)
47 \( 1 + (-0.833 - 0.552i)T \)
53 \( 1 + (0.574 - 0.818i)T \)
59 \( 1 + (-0.530 - 0.847i)T \)
61 \( 1 + (-0.0266 + 0.999i)T \)
67 \( 1 + (-0.132 + 0.991i)T \)
71 \( 1 + (0.484 - 0.874i)T \)
73 \( 1 + (0.388 + 0.921i)T \)
79 \( 1 + (-0.887 + 0.461i)T \)
83 \( 1 + (0.861 + 0.507i)T \)
89 \( 1 + (0.0797 + 0.996i)T \)
97 \( 1 + (-0.769 - 0.638i)T \)
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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

−22.595970753499484001007427018700, −21.63883161648256452903872776724, −21.29799165391338013883786404176, −20.292104329519254968482733986, −19.75254183198300373962710590015, −18.970583126111691786631475477, −18.0687112396510263832901860911, −17.244685804242651556263069818763, −16.11302994289296020858104554784, −15.17896682037656953666634948137, −14.20828746255783679428135789446, −13.74246044621666706966966232683, −12.908338790061342240839723588280, −12.1346862248038524076808186458, −10.65178734839891163202268884112, −10.12269259929942298648896124351, −9.64383940309339056588156778703, −8.86370622575457411552445277437, −7.60312223005525206903602962101, −6.423570824145987013961319341203, −5.16245434916615464947745599193, −4.43250845885288685873692172299, −3.26407920524401133369650439186, −2.54548712860799419690512282828, −1.8143478291746799211783121303, 0.43529889050232892088510405320, 2.27013705867245308667055485483, 2.99855134554455960695557763985, 4.29550044201165589168569184376, 5.449036491608284961848372953454, 6.37008612429861978314631240479, 6.83710624040552757866116295057, 8.18132181096559514555881057477, 8.64454731075469441915026429609, 9.60281730716273745620399363796, 10.248527394648129392406073705667, 12.1752973856250913147911631718, 12.95952872057688032261876704404, 13.347951121585042465776308977089, 14.1091652252706079842170512374, 15.063020545233167930513363170804, 15.752142362133812254817211118537, 16.71624775243837406785786434083, 17.576407124535938843354281736306, 18.20036010496885789997063942483, 19.1654347312349312475819532515, 19.77548634909010417389795962066, 21.051017557395621311708971011230, 21.737591587973482251435451514476, 22.4163825108195779441849691509

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