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Properties

Label 1-709-709.171-r0-0-0
Degree $1$
Conductor $709$
Sign $0.957 + 0.289i$
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  + (−0.833 + 0.552i)2-s + (0.977 − 0.211i)3-s + (0.388 − 0.921i)4-s + (0.994 − 0.106i)5-s + (−0.697 + 0.716i)6-s + (0.574 + 0.818i)7-s + (0.185 + 0.982i)8-s + (0.910 − 0.413i)9-s + (−0.769 + 0.638i)10-s + (−0.964 − 0.263i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (−0.931 − 0.364i)14-s + (0.949 − 0.314i)15-s + (−0.697 − 0.716i)16-s + (−0.964 + 0.263i)17-s + ⋯
L(s)  = 1  + (−0.833 + 0.552i)2-s + (0.977 − 0.211i)3-s + (0.388 − 0.921i)4-s + (0.994 − 0.106i)5-s + (−0.697 + 0.716i)6-s + (0.574 + 0.818i)7-s + (0.185 + 0.982i)8-s + (0.910 − 0.413i)9-s + (−0.769 + 0.638i)10-s + (−0.964 − 0.263i)11-s + (0.185 − 0.982i)12-s + (0.574 − 0.818i)13-s + (−0.931 − 0.364i)14-s + (0.949 − 0.314i)15-s + (−0.697 − 0.716i)16-s + (−0.964 + 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.748417848 + 0.2586267442i\)
\(L(\frac12)\) \(\approx\) \(1.748417848 + 0.2586267442i\)
\(L(1)\) \(\approx\) \(1.253636566 + 0.1727485172i\)
\(L(1)\) \(\approx\) \(1.253636566 + 0.1727485172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.833 + 0.552i)T \)
3 \( 1 + (0.977 - 0.211i)T \)
5 \( 1 + (0.994 - 0.106i)T \)
7 \( 1 + (0.574 + 0.818i)T \)
11 \( 1 + (-0.964 - 0.263i)T \)
13 \( 1 + (0.574 - 0.818i)T \)
17 \( 1 + (-0.964 + 0.263i)T \)
19 \( 1 + (0.185 + 0.982i)T \)
23 \( 1 + (0.658 - 0.752i)T \)
29 \( 1 + (-0.530 - 0.847i)T \)
31 \( 1 + (0.949 - 0.314i)T \)
37 \( 1 + (0.574 + 0.818i)T \)
41 \( 1 + (-0.437 + 0.899i)T \)
43 \( 1 + (-0.998 + 0.0532i)T \)
47 \( 1 + (0.658 - 0.752i)T \)
53 \( 1 + (0.949 - 0.314i)T \)
59 \( 1 + (0.185 + 0.982i)T \)
61 \( 1 + (0.861 + 0.507i)T \)
67 \( 1 + (-0.887 + 0.461i)T \)
71 \( 1 + (-0.769 - 0.638i)T \)
73 \( 1 + (-0.132 - 0.991i)T \)
79 \( 1 + (-0.987 + 0.159i)T \)
83 \( 1 + (-0.339 - 0.940i)T \)
89 \( 1 + (-0.0266 - 0.999i)T \)
97 \( 1 + (0.288 + 0.957i)T \)
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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

−22.0573756011795143084894533679, −21.432228714413421217146155313443, −20.768099843992121087063153973179, −20.29616208552649781436777468998, −19.403203647933925953801749186278, −18.45500525997079256466854439503, −17.86066068873328647541326858138, −17.09403646725738485187116193371, −16.07458147468935479012417117978, −15.30395555324288069777915001745, −14.06980731978105217389729088077, −13.43380705524340917165796205290, −12.91444427212082739455476565320, −11.30457120592770688557699259121, −10.71090371184636176543349607592, −9.9046249909988307309498226536, −9.10061870257734885535663747053, −8.49497948861899228227813323138, −7.30665595152949541956207829788, −6.85970817972618246105395080909, −5.05084923793760667447957213314, −4.09261403459172947616834745270, −2.92316061436642954969092808164, −2.125087325780211820917251341687, −1.274212701635284578790916946145, 1.20330292025527645531005208884, 2.189485339616520331316685009054, 2.84527564005490463136283693066, 4.69479508383690589369803248347, 5.71311496706390310974487861291, 6.39057966355005363881847288669, 7.65033413076334693749315267542, 8.46785985320035703598178147606, 8.7836073255041435276066522352, 9.97851819376324795359219749244, 10.45411414298719709971479928418, 11.71740300642139873335111724088, 13.157668513405214132457526207240, 13.50430411268414311168302836495, 14.782419168473462501409104936211, 15.099772284699118610703764320320, 16.01864714471420960130772688326, 17.0600859330892664986848835483, 18.15405325879532314342645807200, 18.29285020655285147262350337379, 19.10824568354870216435615858408, 20.36710950935148320601254483629, 20.74752799992300302846975965471, 21.51495125782299001424293710228, 22.73956950073946170437666914889

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