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Properties

Label 1-77-77.16-r0-0-0
Degree $1$
Conductor $77$
Sign $0.292 + 0.956i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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  + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.809 + 0.587i)20-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.104 + 0.994i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.104 + 0.994i)16-s + (0.913 − 0.406i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + (−0.809 + 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9442463037 + 0.6988240891i\)
\(L(\frac12)\) \(\approx\) \(0.9442463037 + 0.6988240891i\)
\(L(1)\) \(\approx\) \(1.145030259 + 0.4984948103i\)
\(L(1)\) \(\approx\) \(1.145030259 + 0.4984948103i\)

Euler product

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

−31.172566138919632863802131052335, −29.724528935882130083650445678479, −29.10783989587114791941692530679, −28.02171568619077138431870778504, −27.32550836928007389029831517038, −25.21902605858544259622130303211, −24.1782217905649304909178096730, −23.4194356914808698674278493097, −22.354030028295980456723273760742, −21.37204068580564691611698166245, −20.424165549923272974406697626333, −19.22067600919081805956371587854, −17.59408679068100705174551659813, −16.40142717744921663321117456409, −15.55622856835363975994299013425, −14.05881170721534582972681778596, −12.399734516165618043136858977981, −12.272155974815066278146776938851, −10.69449573325807594986526744791, −9.62162917153890874125358155635, −7.46801752958991146004760676139, −5.73908944736511454230547600460, −5.05093947737247259122393243910, −3.67407271847625370832755303873, −1.342856223193901756105551633436, 2.52494644260805032222452064832, 4.2608404478461868492501779950, 5.61658870394791246801907551998, 6.77279216658514603429071913346, 7.60785484137363936929046827256, 10.08286069603019642299812490505, 11.426217896338131745970801797986, 12.114431459794336638568072070251, 13.57659734434689735545106341451, 14.66974020257679530764687178063, 15.8657410680657203936570854753, 16.92534873997691337898076941141, 18.05475189581830823524962273656, 19.30432021414415948322428523890, 21.048179863264765909209071256346, 22.177620080289219564629280179505, 22.675363067161471006302246352999, 23.7816011759878900136158063233, 24.649418940570576832465766440114, 26.05037964550580471691783358642, 27.021992048998316892867509018714, 28.57414744185050955869543147746, 29.685932271714357201175259891468, 30.29610265405904493749719002058, 31.42933606140196346461261269949

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