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Properties

Label 1-26e2-676.395-r0-0-0
Degree $1$
Conductor $676$
Sign $0.795 - 0.605i$
Analytic cond. $3.13933$
Root an. cond. $3.13933$
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  + (−0.120 − 0.992i)3-s + (0.822 − 0.568i)5-s + (0.663 + 0.748i)7-s + (−0.970 + 0.239i)9-s + (−0.239 + 0.970i)11-s + (−0.663 − 0.748i)15-s + (0.748 − 0.663i)17-s − i·19-s + (0.663 − 0.748i)21-s + 23-s + (0.354 − 0.935i)25-s + (0.354 + 0.935i)27-s + (−0.970 + 0.239i)29-s + (0.935 − 0.354i)31-s + (0.992 + 0.120i)33-s + ⋯
L(s)  = 1  + (−0.120 − 0.992i)3-s + (0.822 − 0.568i)5-s + (0.663 + 0.748i)7-s + (−0.970 + 0.239i)9-s + (−0.239 + 0.970i)11-s + (−0.663 − 0.748i)15-s + (0.748 − 0.663i)17-s − i·19-s + (0.663 − 0.748i)21-s + 23-s + (0.354 − 0.935i)25-s + (0.354 + 0.935i)27-s + (−0.970 + 0.239i)29-s + (0.935 − 0.354i)31-s + (0.992 + 0.120i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.795 - 0.605i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ 0.795 - 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.616291762 - 0.5451690057i\)
\(L(\frac12)\) \(\approx\) \(1.616291762 - 0.5451690057i\)
\(L(1)\) \(\approx\) \(1.226696565 - 0.3126131164i\)
\(L(1)\) \(\approx\) \(1.226696565 - 0.3126131164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.120 - 0.992i)T \)
5 \( 1 + (0.822 - 0.568i)T \)
7 \( 1 + (0.663 + 0.748i)T \)
11 \( 1 + (-0.239 + 0.970i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 - iT \)
23 \( 1 + T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (0.935 - 0.354i)T \)
37 \( 1 + (0.935 - 0.354i)T \)
41 \( 1 + (0.992 - 0.120i)T \)
43 \( 1 + (-0.354 + 0.935i)T \)
47 \( 1 + (0.464 - 0.885i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (0.822 - 0.568i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (-0.464 + 0.885i)T \)
71 \( 1 + (-0.992 + 0.120i)T \)
73 \( 1 + (0.239 - 0.970i)T \)
79 \( 1 + (-0.885 - 0.464i)T \)
83 \( 1 + (0.992 + 0.120i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.822 - 0.568i)T \)
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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

−22.68210617169910669987258459873, −21.86602949017239755742139448668, −21.18149401044510996296941399672, −20.78661328126494674699687981599, −19.626452411394944489532558632421, −18.73780781725597816960622904703, −17.6260660909348929143002043618, −17.1122799883263402679724484988, −16.3985296217354433091426413261, −15.2582009723266484892326854803, −14.59287970526393446657861792723, −13.85220868770460601868613748875, −13.09853956636449584989282150278, −11.52113644441475476685408409448, −10.91037486182208206623473841381, −10.34193461921382830551294457298, −9.41743388377875736451417987416, −8.52403237180080144195829464323, −7.444120004589159702526000852051, −6.25616029090336001200969110825, −5.471059591346238108593526435965, −4.56260262045885471280003706984, −3.46205995024953443643314705666, −2.6220974976884625447805554511, −1.068065878041914833531079692076, 1.16319993823976922929277538258, 1.97433202113738521329006400395, 2.82656673877768743079285891167, 4.64628168674026844548932025755, 5.45856697195303559261022469695, 6.123567177518857722651485417473, 7.35336651026501865341378690742, 8.054205407875339833056312783572, 9.05051581320536961588991051119, 9.829589901271978128141081445227, 11.07700081216566347852483106026, 12.05273748541790062841543218759, 12.59592535625466754780494792087, 13.37078144644380895595259449647, 14.35568445359859450543973671179, 14.94594487008193237580376672662, 16.317145997206194042811014769701, 17.1081357228439025521046255536, 17.86012030818236085526325491879, 18.41913089938345286498532325067, 19.18594670851637864306811303105, 20.522451731565164899592047411488, 20.76202331528080534194090292288, 21.832193263511939130038120718215, 22.843796817222936735399185974835

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