Properties

Degree 1
Conductor $ 13 \cdot 31 $
Sign $-0.957 - 0.289i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  i·2-s − 3-s − 4-s i·5-s + i·6-s + i·7-s + i·8-s + 9-s − 10-s i·11-s + 12-s + 14-s + i·15-s + 16-s + 17-s i·18-s + ⋯
L(s,χ)  = 1  i·2-s − 3-s − 4-s i·5-s + i·6-s + i·7-s + i·8-s + 9-s − 10-s i·11-s + 12-s + 14-s + i·15-s + 16-s + 17-s i·18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.957 - 0.289i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.957 - 0.289i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.957 - 0.289i$
motivic weight  =  \(0\)
character  :  $\chi_{403} (278, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 403,\ (0:\ ),\ -0.957 - 0.289i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1012528031 - 0.6838229577i$
$L(\frac12,\chi)$  $\approx$  $0.1012528031 - 0.6838229577i$
$L(\chi,1)$  $\approx$  0.5241042358 - 0.4669310944i
$L(1,\chi)$  $\approx$  0.5241042358 - 0.4669310944i

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

−24.724445817003609867162021282149, −23.52003979061117265323377266791, −23.11695885838251522235127164234, −22.629587456871983030315261899906, −21.66074374613152535308965007583, −20.55939345498912251349192738437, −19.00098150983091517428471425031, −18.48968059313459771834832567437, −17.46273309808327078680727058342, −16.99027462513987661356771589387, −16.10424235345290806265620653693, −15.030140153314561934318912527456, −14.4195011619579794583436498698, −13.30357876311932408314704995371, −12.413024912628590845449763498294, −11.19392059744044671973163806654, −10.15850854467873741190956969692, −9.73421243965031112367979039109, −7.8250274655449601447148632909, −7.22515808007924694826721263284, −6.5038335472757068225680125534, −5.50631398315045810877498728144, −4.44019932044032077057396136619, −3.495895786214120468193489800812, −1.33048626440042165210630588392, 0.53656517806838309558268311625, 1.66130977585249150463661410726, 3.10684956524393306673994838660, 4.41543562684047598817988973940, 5.37056642362441235448334442569, 5.86621125455799566726753304485, 7.69479331355980497746558203409, 8.94882865230525078270064982005, 9.40541753652437757629472675430, 10.794996754075358869472607871105, 11.43092661381018166065473623275, 12.3578995124874046071480703487, 12.81189953938063274058412901399, 13.8463053083356461989069110186, 15.26096649551305211788900472837, 16.2999801665827858475552542166, 17.07087792795031236275519594815, 17.920474229920446610432364682347, 18.88120406188498712308820792293, 19.41611705928719878485431731858, 20.88242860247361593117578072628, 21.2680177621401668211100942237, 22.080497025711381691093467299988, 22.893024539549357584281127007, 23.95240261573246392957395587381

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