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Properties

Label 1-77-77.26-r1-0-0
Degree $1$
Conductor $77$
Sign $-0.586 - 0.810i$
Analytic cond. $8.27479$
Root an. cond. $8.27479$
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  + (0.669 − 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.669 − 0.743i)17-s + (−0.104 − 0.994i)18-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (−0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (−0.669 − 0.743i)17-s + (−0.104 − 0.994i)18-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.586 - 0.810i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.586 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7736819763 - 1.515170026i\)
\(L(\frac12)\) \(\approx\) \(0.7736819763 - 1.515170026i\)
\(L(1)\) \(\approx\) \(1.030311494 - 0.6985932746i\)
\(L(1)\) \(\approx\) \(1.030311494 - 0.6985932746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.669 - 0.743i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.104 - 0.994i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.104 - 0.994i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−31.41662912914344865145375976788, −30.233367495234399647512353465940, −29.43325120203105852652235515440, −28.41928160929695278199624036435, −26.843902411413797602432095745695, −25.75679354130628031942655045326, −24.6481088670779487579369966453, −23.90101135386405786312603327668, −22.69873433469055811505420842384, −21.89318252843437468427624167929, −21.022330514463974442120039584094, −18.93804745653302096800213506302, −17.6769343379869520300524376266, −17.05771220837937454437859394755, −15.93164068988560249262885086824, −14.411484468662293765018620342901, −13.42080415541261533065836803401, −12.39367156841019988846864067431, −11.14061951266124812113497948716, −9.525448668566125974771280350199, −7.70705969479018772753325266006, −6.41946855837561080809690435167, −5.71890058097564747840369072277, −4.27800707114344239702214413683, −2.04676806342713841879522778570, 0.753916284463055305237652155096, 2.62832271304555309694413566228, 4.54048142083980698752273476317, 5.47333628164352576747290721417, 6.602691621845375806366524676119, 9.25593789725965684167288585463, 10.22660219465915206680515992916, 11.21821181350734695983081027819, 12.51126769771068909210301141006, 13.42258435737894783913978157172, 14.83356356190901978999292362670, 16.06715639868272273161481238065, 17.543749869340090550842111573288, 18.30643353840229513972075901681, 20.06902473950750367380198552490, 20.97670086100806126100640763318, 22.08567227360948118599429646848, 22.5522483830530171308375491073, 23.95568963126336299944382854255, 24.86554865242034349131960103755, 26.61195572979658409140817564860, 27.88508899077680048093039937145, 28.63297288327858779828651374088, 29.52399610904276027003691332126, 30.28835878872832631820884771782

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