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Properties

Label 1-26e2-676.47-r0-0-0
Degree $1$
Conductor $676$
Sign $-0.175 - 0.984i$
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  + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s − i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s − i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(676\)    =    \(2^{2} \cdot 13^{2}\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(3.13933\)
Root analytic conductor: \(3.13933\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{676} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 676,\ (0:\ ),\ -0.175 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6003362776 - 0.7169553146i\)
\(L(\frac12)\) \(\approx\) \(0.6003362776 - 0.7169553146i\)
\(L(1)\) \(\approx\) \(0.7315017283 - 0.3223538015i\)
\(L(1)\) \(\approx\) \(0.7315017283 - 0.3223538015i\)

Euler product

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

−22.89067091228411552516001962554, −22.25158057031116773056695991002, −21.51922864055352068663284799702, −20.816727008888775907448069876171, −19.3510257369414957068745343190, −18.93025263000201176238533534357, −18.15519787112302205167726655619, −17.09478222483286349446421813518, −16.39502717898574402153510755713, −15.53644366696405221351536912602, −14.925890381359816333241555516996, −14.08080551331659637010929548905, −12.58031799543151122989725058063, −11.94180833012883884423814431976, −11.37267386812766901347771107948, −10.419327934620607597661985602381, −9.612199669900761504016109355563, −8.57259567315531574274417471051, −7.50741674128456452812125255915, −6.27136204611452497478621764728, −5.97106779166208142084108631110, −4.67311354288821745639293260850, −3.665623282156886884532442277953, −2.85370802692362787602399456012, −1.116257865844979678858322427452, 0.68610962558817533855120446913, 1.418828151152139105436081783889, 3.20418582751835058582267865769, 4.46465510554987139557577397156, 4.89463794164094366507002406926, 6.244103366292967015106434175128, 7.13706640450052574030294240893, 7.688431746905325337670384847395, 8.93911891251075185328287001983, 9.9342688946259410217557586733, 10.918351278450922548549936489769, 11.71881576960149565945341624320, 12.430194058888386441029695196426, 13.16014103481084625199680541080, 14.04401858470561226607716412295, 15.26384035405670190940315898505, 16.15670273040683097517182665061, 17.01040631590041854989270691501, 17.20502713200000745273391808555, 18.42055167758261468102446747427, 19.38674548865680747183403431124, 19.91392786339231233054087919378, 20.79374596431632758662307097970, 21.8690691328774537958997787791, 22.789988567252438705594645383476

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