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Properties

Label 1-6145-6145.1007-r0-0-0
Degree $1$
Conductor $6145$
Sign $0.563 + 0.825i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
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  + (0.675 − 0.737i)2-s + (−0.953 + 0.302i)3-s + (−0.0868 − 0.996i)4-s + (−0.421 + 0.906i)6-s + (0.694 + 0.719i)7-s + (−0.793 − 0.609i)8-s + (0.817 − 0.576i)9-s + (−0.592 + 0.805i)11-s + (0.383 + 0.923i)12-s + (−0.117 + 0.993i)13-s + (0.999 − 0.0255i)14-s + (−0.984 + 0.173i)16-s + (0.994 − 0.102i)17-s + (0.127 − 0.991i)18-s + (0.369 − 0.929i)19-s + ⋯
L(s)  = 1  + (0.675 − 0.737i)2-s + (−0.953 + 0.302i)3-s + (−0.0868 − 0.996i)4-s + (−0.421 + 0.906i)6-s + (0.694 + 0.719i)7-s + (−0.793 − 0.609i)8-s + (0.817 − 0.576i)9-s + (−0.592 + 0.805i)11-s + (0.383 + 0.923i)12-s + (−0.117 + 0.993i)13-s + (0.999 − 0.0255i)14-s + (−0.984 + 0.173i)16-s + (0.994 − 0.102i)17-s + (0.127 − 0.991i)18-s + (0.369 − 0.929i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.563 + 0.825i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ 0.563 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257823612 + 0.6643465070i\)
\(L(\frac12)\) \(\approx\) \(1.257823612 + 0.6643465070i\)
\(L(1)\) \(\approx\) \(1.084070606 - 0.1626697351i\)
\(L(1)\) \(\approx\) \(1.084070606 - 0.1626697351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (0.675 - 0.737i)T \)
3 \( 1 + (-0.953 + 0.302i)T \)
7 \( 1 + (0.694 + 0.719i)T \)
11 \( 1 + (-0.592 + 0.805i)T \)
13 \( 1 + (-0.117 + 0.993i)T \)
17 \( 1 + (0.994 - 0.102i)T \)
19 \( 1 + (0.369 - 0.929i)T \)
23 \( 1 + (-0.956 - 0.292i)T \)
29 \( 1 + (0.567 + 0.823i)T \)
31 \( 1 + (0.999 + 0.0409i)T \)
37 \( 1 + (-0.946 - 0.321i)T \)
41 \( 1 + (0.364 + 0.931i)T \)
43 \( 1 + (0.744 + 0.668i)T \)
47 \( 1 + (-0.998 + 0.0562i)T \)
53 \( 1 + (-0.927 - 0.374i)T \)
59 \( 1 + (0.983 + 0.183i)T \)
61 \( 1 + (0.448 + 0.893i)T \)
67 \( 1 + (-0.985 - 0.168i)T \)
71 \( 1 + (-0.0919 - 0.995i)T \)
73 \( 1 + (0.957 - 0.287i)T \)
79 \( 1 + (-0.793 + 0.609i)T \)
83 \( 1 + (0.981 + 0.193i)T \)
89 \( 1 + (-0.0766 - 0.997i)T \)
97 \( 1 + (0.435 - 0.900i)T \)
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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

−17.43396523085177131279777555757, −16.99880783890510105128233950294, −16.01707817129238113219078651955, −15.9663910537819438981202134079, −14.99219717569015940722306539898, −14.05835538502103457154010523998, −13.817689066875419438706860596078, −13.04780399996632585326489014006, −12.290220663796544477998051679221, −11.87799890266031408987010009066, −11.115152194506670028358428677040, −10.35191682160008202888248856193, −9.907386800545748363479245473, −8.40881317434702711520400657648, −7.88933816230474577254443897895, −7.63104531681381486549009741384, −6.67333089409524473459762045971, −5.91328398269242628799285499137, −5.46243841259996301720329874009, −4.9195331052434017541044958160, −4.00377427596107968202207715943, −3.41221022496034867818085243707, −2.37623514719712399811761282851, −1.27402169158581324222460512670, −0.35934546317284591974314875020, 1.02323730364113882259367644448, 1.748918711232539959487903865039, 2.51956354224896341884169074917, 3.37024612279674533626046301569, 4.48915145396029118513817623647, 4.71795586120714863934764165808, 5.34716504581673830133871271728, 6.09240645639103614018892896904, 6.75892204461836178583127976138, 7.58960444395279222195762890801, 8.63577067326540441722755280260, 9.52349525461983492302957783322, 9.92452968112043653706841862657, 10.68561410400574837972239047810, 11.33601301641933544407813227557, 11.93522321536349502129551423567, 12.26215583329767049683084866600, 12.94085917504255484349064207962, 13.85538006216048229436076909976, 14.54355693164025516940010414955, 15.02166204594235580919591203551, 15.9729639185794486755054390050, 16.08910210778280027548843144217, 17.305424325432469632027848273857, 18.08049012151832151859892227751

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