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Properties

Label 1-2557-2557.4-r0-0-0
Degree $1$
Conductor $2557$
Sign $-0.690 - 0.723i$
Analytic cond. $11.8746$
Root an. cond. $11.8746$
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  + (0.999 + 0.00491i)2-s + (0.648 − 0.761i)3-s + (0.999 + 0.00983i)4-s + (0.614 − 0.789i)5-s + (0.652 − 0.758i)6-s + (−0.817 − 0.576i)7-s + (0.999 + 0.0147i)8-s + (−0.159 − 0.987i)9-s + (0.617 − 0.786i)10-s + (0.134 − 0.990i)11-s + (0.655 − 0.754i)12-s + (−0.985 − 0.171i)13-s + (−0.814 − 0.580i)14-s + (−0.202 − 0.979i)15-s + (0.999 + 0.0196i)16-s + (0.988 + 0.151i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00491i)2-s + (0.648 − 0.761i)3-s + (0.999 + 0.00983i)4-s + (0.614 − 0.789i)5-s + (0.652 − 0.758i)6-s + (−0.817 − 0.576i)7-s + (0.999 + 0.0147i)8-s + (−0.159 − 0.987i)9-s + (0.617 − 0.786i)10-s + (0.134 − 0.990i)11-s + (0.655 − 0.754i)12-s + (−0.985 − 0.171i)13-s + (−0.814 − 0.580i)14-s + (−0.202 − 0.979i)15-s + (0.999 + 0.0196i)16-s + (0.988 + 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $-0.690 - 0.723i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ -0.690 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.668264810 - 3.896881111i\)
\(L(\frac12)\) \(\approx\) \(1.668264810 - 3.896881111i\)
\(L(1)\) \(\approx\) \(2.004164902 - 1.392826673i\)
\(L(1)\) \(\approx\) \(2.004164902 - 1.392826673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.999 + 0.00491i)T \)
3 \( 1 + (0.648 - 0.761i)T \)
5 \( 1 + (0.614 - 0.789i)T \)
7 \( 1 + (-0.817 - 0.576i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (-0.985 - 0.171i)T \)
17 \( 1 + (0.988 + 0.151i)T \)
19 \( 1 + (-0.929 - 0.369i)T \)
23 \( 1 + (0.967 - 0.252i)T \)
29 \( 1 + (0.872 + 0.489i)T \)
31 \( 1 + (-0.997 - 0.0687i)T \)
37 \( 1 + (0.425 + 0.904i)T \)
41 \( 1 + (0.274 + 0.961i)T \)
43 \( 1 + (0.0614 + 0.998i)T \)
47 \( 1 + (0.985 - 0.171i)T \)
53 \( 1 + (-0.525 - 0.850i)T \)
59 \( 1 + (-0.743 - 0.668i)T \)
61 \( 1 + (-0.787 + 0.616i)T \)
67 \( 1 + (0.245 + 0.969i)T \)
71 \( 1 + (-0.149 + 0.988i)T \)
73 \( 1 + (-0.955 - 0.295i)T \)
79 \( 1 + (0.883 + 0.467i)T \)
83 \( 1 + (-0.998 + 0.0491i)T \)
89 \( 1 + (0.207 - 0.978i)T \)
97 \( 1 + (0.890 - 0.454i)T \)
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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

−19.67239252023361582349264703524, −19.2488243596833274607405707377, −18.51268786382191993332060186314, −17.12960400129567420976092041449, −16.8338778370091069897361502976, −15.70678061241716574551243863061, −15.2591865677899005713211552410, −14.60533010407539572210898936986, −14.21763487534989042231401365092, −13.35638500902341342612022373195, −12.5162043096290098090112337118, −12.064933131414859760024022125188, −10.77871649379823059733516668489, −10.38132904216408443804879128247, −9.58611126037841920107630414710, −9.0684676326784875717749720340, −7.594386823175012904747993206149, −7.19629885717870946301821732879, −6.19584350296141030901238224715, −5.50587041045922351759549979420, −4.731724125304093949687138158203, −3.83839137592894868791043028570, −3.088507111505972388808206904332, −2.42340012278757643420062416882, −1.89154013408296667953865415499, 0.79149579181814517395364459192, 1.52690364603584035618325531047, 2.73205036402955651432583527744, 3.07477180833036111540537423887, 4.11782831089924712481882237106, 4.97162151152862919838547726137, 5.931615409208131142406732385452, 6.45025061279793367793526569062, 7.23144218866154252699715865678, 8.03359812836392116925257793793, 8.85763343825434379481793204834, 9.722012509311931843926164924069, 10.45507178762804994044777863831, 11.49332702234859299158514458976, 12.38698049849266091492096450599, 12.96705921401333495949049764852, 13.16328990116521928095937608719, 14.17840667017442629051034531067, 14.458093593206895717672421279511, 15.41131579097921835493652685213, 16.46284164734002789066604717642, 16.78878973852052001785924863466, 17.50944672731212095102155497562, 18.769126548860940978121882486711, 19.36445207264607870798680219530

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