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Properties

Label 1-1425-1425.1223-r0-0-0
Degree $1$
Conductor $1425$
Sign $-0.121 + 0.992i$
Analytic cond. $6.61767$
Root an. cond. $6.61767$
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  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s − i·7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)11-s + (0.743 + 0.669i)13-s + (0.669 + 0.743i)14-s + (−0.978 − 0.207i)16-s + (0.994 − 0.104i)17-s + (−0.406 − 0.913i)22-s + (0.207 + 0.978i)23-s − 26-s + (−0.994 − 0.104i)28-s + (−0.104 + 0.994i)29-s + (−0.809 + 0.587i)31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s − i·7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)11-s + (0.743 + 0.669i)13-s + (0.669 + 0.743i)14-s + (−0.978 − 0.207i)16-s + (0.994 − 0.104i)17-s + (−0.406 − 0.913i)22-s + (0.207 + 0.978i)23-s − 26-s + (−0.994 − 0.104i)28-s + (−0.104 + 0.994i)29-s + (−0.809 + 0.587i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(6.61767\)
Root analytic conductor: \(6.61767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1425,\ (0:\ ),\ -0.121 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6150325063 + 0.6952216772i\)
\(L(\frac12)\) \(\approx\) \(0.6150325063 + 0.6952216772i\)
\(L(1)\) \(\approx\) \(0.7168110170 + 0.2566077251i\)
\(L(1)\) \(\approx\) \(0.7168110170 + 0.2566077251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.743 + 0.669i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (0.994 - 0.104i)T \)
23 \( 1 + (0.207 + 0.978i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (0.994 + 0.104i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−20.643360040400901663954654070259, −19.68120408620036804925949298418, −18.8629211747517424273513113548, −18.47346133797948090732952449479, −17.844704965425534693016614991790, −16.71249313243738353948676511198, −16.28129989875573329999790747127, −15.397179364968382513873051437306, −14.54065175529566968571001789946, −13.32577057368793846124294940945, −12.856991405531744947399997537182, −11.92880424661041172692356808299, −11.29277375178710670950076108894, −10.54419188830381004709382418737, −9.71739138600898127198164114301, −8.865949471805621724032544653757, −8.23148065637514619761427292582, −7.63039687717960599144128409536, −6.23618548207897558507103119793, −5.67651678878346068962523093989, −4.40073528392637681218071187605, −3.25148715697871139840313299680, −2.773529167108616970852081856595, −1.64432976671821253101590027828, −0.52018927228317841840338201292, 1.1246453511842067912443370772, 1.79980962895171833712744551267, 3.30044433080589387008821824997, 4.35325628247406414160281483739, 5.20731151652979484092488881675, 6.13209306249581232000818343592, 7.18611588401754090095092292018, 7.417763124182927768612319779725, 8.43975579321597839555979251413, 9.38342126613839322615685544886, 9.97908324691196523907720752375, 10.78293065545360867508747162094, 11.45819518772377479797353660729, 12.64987602002270452178993850350, 13.553880646574373969789014221224, 14.284309254600435364834819958933, 14.92382395824280621686274437867, 15.87666310364744121507421591945, 16.530493009566094871709712589609, 17.05813643913100390400221527494, 18.06243456672679835965052583586, 18.374230351040569400569489796812, 19.505721861380753676445130704864, 19.987707829882002792424247011382, 20.76618099024758970598614077398

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