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Properties

Label 1-697-697.696-r0-0-0
Degree $1$
Conductor $697$
Sign $1$
Analytic cond. $3.23685$
Root an. cond. $3.23685$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 18-s − 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + 29-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 18-s − 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 697 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(697\)    =    \(17 \cdot 41\)
Sign: $1$
Analytic conductor: \(3.23685\)
Root analytic conductor: \(3.23685\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{697} (696, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 697,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.769267311\)
\(L(\frac12)\) \(\approx\) \(3.769267311\)
\(L(1)\) \(\approx\) \(2.534456113\)
\(L(1)\) \(\approx\) \(2.534456113\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
41 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−22.56234978802482841929789820033, −21.8150637407204786883316331426, −21.04421626074274458270101028823, −20.2241279749667385714162667751, −19.61225212092448026549558870518, −19.10362763654379459247855383503, −17.71752301438053940164810673359, −16.64251423679064034722504681952, −15.747673102995701157965844649278, −14.87898284186516377874635131854, −14.537352493440427577241459556321, −13.84349272037195230190306040397, −12.54493572350382632763012209630, −12.107614576816095174047527500550, −11.161571905712254791736857852170, −10.23167680545347048201749920763, −8.89547169056288089588298755339, −8.0140418832339728247930768698, −7.37170538274121869401486284453, −6.47848953018578547207765721151, −4.927742351846481205736946004233, −4.27605806848481860734626132916, −3.58273578481851579180544562064, −2.42445411913704695342458409237, −1.51933619136360730719844363597, 1.51933619136360730719844363597, 2.42445411913704695342458409237, 3.58273578481851579180544562064, 4.27605806848481860734626132916, 4.927742351846481205736946004233, 6.47848953018578547207765721151, 7.37170538274121869401486284453, 8.0140418832339728247930768698, 8.89547169056288089588298755339, 10.23167680545347048201749920763, 11.161571905712254791736857852170, 12.107614576816095174047527500550, 12.54493572350382632763012209630, 13.84349272037195230190306040397, 14.537352493440427577241459556321, 14.87898284186516377874635131854, 15.747673102995701157965844649278, 16.64251423679064034722504681952, 17.71752301438053940164810673359, 19.10362763654379459247855383503, 19.61225212092448026549558870518, 20.2241279749667385714162667751, 21.04421626074274458270101028823, 21.8150637407204786883316331426, 22.56234978802482841929789820033

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