Properties

Degree 1
Conductor $ 3 \cdot 5 \cdot 61 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s + 7-s − 8-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s − 22-s − 23-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 37-s − 38-s − 41-s + 43-s + 44-s + 46-s + 47-s + 49-s − 52-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s + 7-s − 8-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s − 22-s − 23-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 37-s − 38-s − 41-s + 43-s + 44-s + 46-s + 47-s + 49-s − 52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 915 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(915\)    =    \(3 \cdot 5 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{915} (914, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 915,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.502485849$
$L(\frac12,\chi)$  $\approx$  $1.502485849$
$L(\chi,1)$  $\approx$  0.8308627957
$L(1,\chi)$  $\approx$  0.8308627957

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.88747701156733947131509334682, −20.514091198315445785695201585361, −20.10454443235284857078474313381, −19.3895674973801592785290431169, −18.37051382159256829118688765395, −17.69404218749881731752412830174, −17.205834907490256822865693557604, −16.29019070467204559789517117618, −15.44800858088245977230532958259, −14.5769494021970679135224801728, −13.98460816568452744141407167655, −12.50583386417821603904005829140, −11.73944858475573174253795036879, −11.21293587407372581573785043495, −10.19136630112533482028006377619, −9.38864283529383548795238493151, −8.6413454681331202733070654505, −7.739398035189489556986981430692, −7.056806754791235581170233137078, −6.08313785554983175136294898810, −4.98600623201739455607424924295, −3.91005468719411936648766788298, −2.54565859463529656799956813083, −1.73375720416775968670569766990, −0.68862504180570466637659808733, 0.68862504180570466637659808733, 1.73375720416775968670569766990, 2.54565859463529656799956813083, 3.91005468719411936648766788298, 4.98600623201739455607424924295, 6.08313785554983175136294898810, 7.056806754791235581170233137078, 7.739398035189489556986981430692, 8.6413454681331202733070654505, 9.38864283529383548795238493151, 10.19136630112533482028006377619, 11.21293587407372581573785043495, 11.73944858475573174253795036879, 12.50583386417821603904005829140, 13.98460816568452744141407167655, 14.5769494021970679135224801728, 15.44800858088245977230532958259, 16.29019070467204559789517117618, 17.205834907490256822865693557604, 17.69404218749881731752412830174, 18.37051382159256829118688765395, 19.3895674973801592785290431169, 20.10454443235284857078474313381, 20.514091198315445785695201585361, 21.88747701156733947131509334682

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