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Properties

Label 1-709-709.175-r0-0-0
Degree $1$
Conductor $709$
Sign $0.954 + 0.297i$
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  + (0.994 − 0.106i)2-s + (0.861 + 0.507i)3-s + (0.977 − 0.211i)4-s + (−0.964 − 0.263i)5-s + (0.910 + 0.413i)6-s + (0.734 + 0.678i)7-s + (0.949 − 0.314i)8-s + (0.484 + 0.874i)9-s + (−0.987 − 0.159i)10-s + (−0.617 − 0.786i)11-s + (0.949 + 0.314i)12-s + (0.734 − 0.678i)13-s + (0.802 + 0.596i)14-s + (−0.697 − 0.716i)15-s + (0.910 − 0.413i)16-s + (−0.617 + 0.786i)17-s + ⋯
L(s)  = 1  + (0.994 − 0.106i)2-s + (0.861 + 0.507i)3-s + (0.977 − 0.211i)4-s + (−0.964 − 0.263i)5-s + (0.910 + 0.413i)6-s + (0.734 + 0.678i)7-s + (0.949 − 0.314i)8-s + (0.484 + 0.874i)9-s + (−0.987 − 0.159i)10-s + (−0.617 − 0.786i)11-s + (0.949 + 0.314i)12-s + (0.734 − 0.678i)13-s + (0.802 + 0.596i)14-s + (−0.697 − 0.716i)15-s + (0.910 − 0.413i)16-s + (−0.617 + 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.954 + 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.450471084 + 0.5251081692i\)
\(L(\frac12)\) \(\approx\) \(3.450471084 + 0.5251081692i\)
\(L(1)\) \(\approx\) \(2.360855554 + 0.2117193519i\)
\(L(1)\) \(\approx\) \(2.360855554 + 0.2117193519i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.994 - 0.106i)T \)
3 \( 1 + (0.861 + 0.507i)T \)
5 \( 1 + (-0.964 - 0.263i)T \)
7 \( 1 + (0.734 + 0.678i)T \)
11 \( 1 + (-0.617 - 0.786i)T \)
13 \( 1 + (0.734 - 0.678i)T \)
17 \( 1 + (-0.617 + 0.786i)T \)
19 \( 1 + (0.949 - 0.314i)T \)
23 \( 1 + (-0.530 + 0.847i)T \)
29 \( 1 + (0.574 - 0.818i)T \)
31 \( 1 + (-0.697 - 0.716i)T \)
37 \( 1 + (0.734 + 0.678i)T \)
41 \( 1 + (-0.339 - 0.940i)T \)
43 \( 1 + (-0.132 + 0.991i)T \)
47 \( 1 + (-0.530 + 0.847i)T \)
53 \( 1 + (-0.697 - 0.716i)T \)
59 \( 1 + (0.949 - 0.314i)T \)
61 \( 1 + (-0.237 + 0.971i)T \)
67 \( 1 + (-0.931 + 0.364i)T \)
71 \( 1 + (-0.987 + 0.159i)T \)
73 \( 1 + (-0.437 - 0.899i)T \)
79 \( 1 + (0.388 - 0.921i)T \)
83 \( 1 + (0.0797 - 0.996i)T \)
89 \( 1 + (0.658 + 0.752i)T \)
97 \( 1 + (-0.998 + 0.0532i)T \)
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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

−22.86911701742026883718567658550, −21.73557230830533712938825686357, −20.68895631699788261523647566158, −20.26437797264118375210625296332, −19.76599282837962607262197617749, −18.45341493538663614209433756836, −17.99153112513264876424039992969, −16.405853805122992148110963489102, −15.83364181673492149149872705010, −14.92040772502241861929351830145, −14.26607879175798529230230429132, −13.67808760389141559638162886822, −12.71083237010158996984069552279, −11.92179201623949261499263798038, −11.15701936151937606335271881157, −10.20968268606427460468256389609, −8.7146963452285929487838170875, −7.83416859924602969280396089057, −7.223828214110554707789345200009, −6.607431130219491495333613243380, −4.9786879953212383736649055473, −4.22120806555617361016297854408, −3.430994626459933627290436753852, −2.41267839526935137946420825656, −1.34573532046907936351685647680, 1.46939269230958757740235025794, 2.71665790769021293441037458694, 3.44644845681908479914107673945, 4.31574909701421491517535338473, 5.16862118366599909760590869305, 6.044153893007401788798707879151, 7.706294837305243272976281155853, 8.00994664612500591803581331549, 8.97018668834808700011587043394, 10.318186341337385084021515856574, 11.21132933240539540671212181775, 11.709780492092465143624928648433, 13.01132690472463605291058574168, 13.461634342237556525333251615367, 14.5511117307808833947695663301, 15.2555550569996333723554995572, 15.76500982551339622391633977995, 16.32358263161678691996897202355, 17.86142357658187387537063009023, 19.00062088347972952921712247286, 19.58865820963130343858014413689, 20.53808936431735287664476384759, 20.88765414404797011895075478334, 21.86262354706469901071160546140, 22.39127547626760729827430037778

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