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Properties

Label 1-26e2-676.115-r0-0-0
Degree $1$
Conductor $676$
Sign $0.999 + 0.00929i$
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  + (0.996 − 0.0804i)3-s + (0.992 + 0.120i)5-s + (0.534 − 0.845i)7-s + (0.987 − 0.160i)9-s + (−0.160 + 0.987i)11-s + (0.999 + 0.0402i)15-s + (0.845 + 0.534i)17-s + (−0.866 + 0.5i)19-s + (0.464 − 0.885i)21-s + (−0.5 + 0.866i)23-s + (0.970 + 0.239i)25-s + (0.970 − 0.239i)27-s + (−0.632 − 0.774i)29-s + (0.239 + 0.970i)31-s + (−0.0804 + 0.996i)33-s + ⋯
L(s)  = 1  + (0.996 − 0.0804i)3-s + (0.992 + 0.120i)5-s + (0.534 − 0.845i)7-s + (0.987 − 0.160i)9-s + (−0.160 + 0.987i)11-s + (0.999 + 0.0402i)15-s + (0.845 + 0.534i)17-s + (−0.866 + 0.5i)19-s + (0.464 − 0.885i)21-s + (−0.5 + 0.866i)23-s + (0.970 + 0.239i)25-s + (0.970 − 0.239i)27-s + (−0.632 − 0.774i)29-s + (0.239 + 0.970i)31-s + (−0.0804 + 0.996i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $0.999 + 0.00929i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ 0.999 + 0.00929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.601882175 + 0.01209188280i\)
\(L(\frac12)\) \(\approx\) \(2.601882175 + 0.01209188280i\)
\(L(1)\) \(\approx\) \(1.802789351 + 0.01957337311i\)
\(L(1)\) \(\approx\) \(1.802789351 + 0.01957337311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.996 - 0.0804i)T \)
5 \( 1 + (0.992 + 0.120i)T \)
7 \( 1 + (0.534 - 0.845i)T \)
11 \( 1 + (-0.160 + 0.987i)T \)
17 \( 1 + (0.845 + 0.534i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.632 - 0.774i)T \)
31 \( 1 + (0.239 + 0.970i)T \)
37 \( 1 + (-0.960 - 0.278i)T \)
41 \( 1 + (-0.0804 - 0.996i)T \)
43 \( 1 + (0.278 + 0.960i)T \)
47 \( 1 + (-0.663 - 0.748i)T \)
53 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (-0.391 - 0.919i)T \)
61 \( 1 + (-0.0402 - 0.999i)T \)
67 \( 1 + (-0.979 + 0.200i)T \)
71 \( 1 + (-0.903 - 0.428i)T \)
73 \( 1 + (-0.935 + 0.354i)T \)
79 \( 1 + (0.748 - 0.663i)T \)
83 \( 1 + (-0.822 - 0.568i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.600 + 0.799i)T \)
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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

−22.41198259856927560549246079311, −21.71840004538181462352704347885, −21.042819081431949350040879803629, −20.607938857715427101295650563657, −19.40019898341646844440262629797, −18.59538017705896049817580279235, −18.14386673421359605995339259343, −16.91366040917124681518148460558, −16.14681808831014799194102588210, −15.06783845891943231434428287587, −14.45199815304961112687245512930, −13.6936957559512580038160594753, −12.96673220825601679213476648087, −11.9955177108837687063437575411, −10.7860124809333123243044384049, −9.96698355006616263481926355623, −8.92427608199666770685638073525, −8.59613915953055343513716230709, −7.51609878811565337259046689923, −6.26680351238929396589976237791, −5.42916841623183154283400178160, −4.42610807640367083034358144409, −3.0400284386303957517533035744, −2.38231900514296780293793287927, −1.35890526803457731785704632340, 1.56112695925019595303493767232, 1.991520104977219836133374577830, 3.36393693651655817469332012679, 4.26401557263845198480352533298, 5.33643084471594602658497860411, 6.572358705805873411707211473744, 7.45258183992339363332444957285, 8.18696195944677110143593344986, 9.26637993611119165422018063835, 10.13350729473012153601312234795, 10.52607494850528096625168483462, 12.08482465572677727033706805982, 13.00302338526458588454848763650, 13.67271354744653647257662258052, 14.494408919489561863450440057882, 14.92859738846705464440215377727, 16.13368403397237743675675309122, 17.27971128114218614847860263793, 17.698982320007686588017318719870, 18.735444145163220785892353257776, 19.55792600465249939687336790396, 20.45490134618857774273817122658, 21.05386041563429412874193519572, 21.555541587767115856969752473600, 22.85066260617032419324221063205

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