Properties

Degree 1
Conductor 5
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 − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{5} (4, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 5,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2317509475$
$L(\frac12,\chi)$  $\approx$  $0.2317509475$
$L(\chi,1)$  $\approx$  0.4304089409
$L(1,\chi)$  $\approx$  0.4304089409

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

−55.58928033540481015096458899849, −53.83044519544216335105426902233, −52.1259022313169741886041306959, −51.08775192674649135525825720413, −48.34566182106784617654820130449, −46.49272715949140534533919935167, −45.4273000827822893888415619096, −44.03129006144169504470090805842, −41.84243854579169430850930688531, −39.56057294640318170505509505995, −38.12918472143653185015141827037, −35.86863837181227459459504863887, −34.728812978904808674143729833981, −33.00045600687051436794975917721, −29.70790935048096556923098651865, −28.46103510017752247518697827232, −26.77609594800414011652357496527, −24.58846621740819520765626997608, −22.227405454459410911877624963081, −19.54073262278475025037869002299, −17.566994292325555202701595268144, −16.03382112838423567459325378224, −11.95884562608351453026565868826, −9.831444432886669616348321347458, −6.64845334472771471612327845997, 6.64845334472771471612327845997, 9.831444432886669616348321347458, 11.95884562608351453026565868826, 16.03382112838423567459325378224, 17.566994292325555202701595268144, 19.54073262278475025037869002299, 22.227405454459410911877624963081, 24.58846621740819520765626997608, 26.77609594800414011652357496527, 28.46103510017752247518697827232, 29.70790935048096556923098651865, 33.00045600687051436794975917721, 34.728812978904808674143729833981, 35.86863837181227459459504863887, 38.12918472143653185015141827037, 39.56057294640318170505509505995, 41.84243854579169430850930688531, 44.03129006144169504470090805842, 45.4273000827822893888415619096, 46.49272715949140534533919935167, 48.34566182106784617654820130449, 51.08775192674649135525825720413, 52.1259022313169741886041306959, 53.83044519544216335105426902233, 55.58928033540481015096458899849

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