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Properties

Label 1-77-77.58-r0-0-0
Degree $1$
Conductor $77$
Sign $0.0395 - 0.999i$
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  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)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 + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (0.309 + 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)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 + (0.669 − 0.743i)16-s + (−0.978 − 0.207i)17-s + (0.913 − 0.406i)18-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0395 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0395 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.0395 - 0.999i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.0395 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4589780525 - 0.4411525607i\)
\(L(\frac12)\) \(\approx\) \(0.4589780525 - 0.4411525607i\)
\(L(1)\) \(\approx\) \(0.6476723973 - 0.2985769157i\)
\(L(1)\) \(\approx\) \(0.6476723973 - 0.2985769157i\)

Euler product

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

−31.42419950625557560991030902424, −30.2391149190694466077968694706, −29.01267597237802035788253517126, −28.35801267427729113521190655322, −27.10199320863722619690751530118, −26.25391128660214742663111631951, −25.701118457208868194290113099007, −24.18882937147961157412626360937, −22.39957849752455733144768041223, −21.599965217010592177804059128170, −20.643822960809841932325042757904, −19.44413321996789631775185117384, −18.13955308679429178660593983553, −17.27024389270937391982824799106, −16.10847936282743548538623755600, −15.10383346502125011903092665089, −13.72353871669230264298217521483, −11.636421085167728843044671405497, −10.82159132447325427472777142582, −9.7072349491986737477046949939, −8.90566240860414258671071025466, −7.10214338142543931851792881622, −5.790514889983247207780073431035, −3.72115519101261125281382675087, −2.2232882267185419524740772252, 1.05579301589600281019069120816, 2.47472211027939259189002604237, 5.4481447631619617913231806570, 6.50409605319997825251682704990, 7.91337423597837233491926456012, 8.84584975415468956753847568504, 10.224605654782553723421533785598, 11.66547160483855585945065247982, 12.8556229474564342447183662949, 14.05891444639369134962844692371, 15.72335685613782531630199182112, 16.959757231466721125116036703821, 17.789116798187120151436878795, 18.59661398109408630387540607295, 20.00897997198843423262442436068, 20.61165175582164104614533120, 22.46600825055167831486939390296, 23.96083523785999049294583319083, 24.739385632662065230234680828957, 25.38730309718046578269528442829, 26.62721483633812466652959319854, 28.08012096265825180089638759401, 28.777815273627585115605938447993, 29.62669086965922331256430140957, 30.67659348172208514293503444924

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