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Properties

Label 1-26e2-676.503-r1-0-0
Degree $1$
Conductor $676$
Sign $-0.764 - 0.644i$
Analytic cond. $72.6462$
Root an. cond. $72.6462$
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  + (0.845 + 0.534i)3-s + (−0.748 − 0.663i)5-s + (−0.692 + 0.721i)7-s + (0.428 + 0.903i)9-s + (−0.428 + 0.903i)11-s + (−0.278 − 0.960i)15-s + (0.692 − 0.721i)17-s + (0.5 − 0.866i)19-s + (−0.970 + 0.239i)21-s + (0.5 + 0.866i)23-s + (0.120 + 0.992i)25-s + (−0.120 + 0.992i)27-s + (−0.996 − 0.0804i)29-s + (−0.120 + 0.992i)31-s + (−0.845 + 0.534i)33-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)3-s + (−0.748 − 0.663i)5-s + (−0.692 + 0.721i)7-s + (0.428 + 0.903i)9-s + (−0.428 + 0.903i)11-s + (−0.278 − 0.960i)15-s + (0.692 − 0.721i)17-s + (0.5 − 0.866i)19-s + (−0.970 + 0.239i)21-s + (0.5 + 0.866i)23-s + (0.120 + 0.992i)25-s + (−0.120 + 0.992i)27-s + (−0.996 − 0.0804i)29-s + (−0.120 + 0.992i)31-s + (−0.845 + 0.534i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.764 - 0.644i$
Analytic conductor: \(72.6462\)
Root analytic conductor: \(72.6462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (1:\ ),\ -0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03272343764 + 0.08961882419i\)
\(L(\frac12)\) \(\approx\) \(-0.03272343764 + 0.08961882419i\)
\(L(1)\) \(\approx\) \(0.9329672295 + 0.2220567248i\)
\(L(1)\) \(\approx\) \(0.9329672295 + 0.2220567248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.845 + 0.534i)T \)
5 \( 1 + (-0.748 - 0.663i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (-0.428 + 0.903i)T \)
17 \( 1 + (0.692 - 0.721i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (-0.120 + 0.992i)T \)
37 \( 1 + (-0.919 + 0.391i)T \)
41 \( 1 + (-0.845 - 0.534i)T \)
43 \( 1 + (0.919 + 0.391i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.948 + 0.316i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.987 + 0.160i)T \)
71 \( 1 + (0.0402 - 0.999i)T \)
73 \( 1 + (0.568 + 0.822i)T \)
79 \( 1 + (0.354 - 0.935i)T \)
83 \( 1 + (-0.885 - 0.464i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.200 - 0.979i)T \)
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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

−22.21890293363895412552917514101, −20.947654408423267393223661228454, −20.37790812190198407301488595226, −19.366078659145211694033385127813, −18.926837578815403963616187139673, −18.388036809137983104229654444413, −17.01606256877903261443594147955, −16.208364145622730243262058441, −15.319057072835966927233044912005, −14.462191873911976225534410333045, −13.83648516301620731806162838768, −12.899444398732788823148969226942, −12.19589600939786926842940056445, −10.9898438285667500074559648464, −10.27102157133110888698910995798, −9.23648238732544267928944019872, −8.09544756337235057616343951894, −7.62518880053486307618664909707, −6.70749821660699219999332918346, −5.82779984228670681894040256121, −4.043370684116152815260892698894, −3.45519874188029418468704405254, −2.68409694600017319083412533359, −1.188875756302720520360150913550, −0.020360257433548018631598294825, 1.65687887838591162577992582984, 2.92122166995284303366080359040, 3.57333256010536352579880775513, 4.83712938763529761490498127651, 5.338936835460325483395928630837, 7.097182731423734211949672034589, 7.68571434936476192513805571383, 8.84304435881856623366648853911, 9.311446052338195945247409539139, 10.15049830632808556907920671609, 11.38033538450211920698975713370, 12.31325670201449811743663748655, 13.01513810611604948814048536576, 13.90872147072636459656954632203, 15.10899191563333968437195788132, 15.56628752058614553178266121514, 16.12701808073886902169049757138, 17.123750066132190636444596152073, 18.38413642298681663838458090887, 19.153041044223323603645029874510, 19.85380792290828737055773586839, 20.54182365364614683132099865816, 21.22523030479485741389696593315, 22.219702605737039071267061345663, 22.94730571540611296192509677405

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