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Properties

Label 1-776-776.131-r0-0-0
Degree $1$
Conductor $776$
Sign $0.944 - 0.329i$
Analytic cond. $3.60372$
Root an. cond. $3.60372$
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  + (−0.382 − 0.923i)3-s + (0.195 − 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.980 − 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (0.195 − 0.980i)19-s + (0.831 − 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (0.923 + 0.382i)27-s + (0.555 + 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.195 − 0.980i)5-s + (0.195 + 0.980i)7-s + (−0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.980 − 0.195i)13-s + (−0.980 + 0.195i)15-s + (0.980 + 0.195i)17-s + (0.195 − 0.980i)19-s + (0.831 − 0.555i)21-s + (0.555 + 0.831i)23-s + (−0.923 − 0.382i)25-s + (0.923 + 0.382i)27-s + (0.555 + 0.831i)29-s + (−0.923 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(3.60372\)
Root analytic conductor: \(3.60372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 776,\ (0:\ ),\ 0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248631760 - 0.2115862534i\)
\(L(\frac12)\) \(\approx\) \(1.248631760 - 0.2115862534i\)
\(L(1)\) \(\approx\) \(0.9788336983 - 0.2135558211i\)
\(L(1)\) \(\approx\) \(0.9788336983 - 0.2135558211i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (0.195 - 0.980i)T \)
7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (0.980 + 0.195i)T \)
19 \( 1 + (0.195 - 0.980i)T \)
23 \( 1 + (0.555 + 0.831i)T \)
29 \( 1 + (0.555 + 0.831i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (0.831 + 0.555i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (0.555 - 0.831i)T \)
61 \( 1 - T \)
67 \( 1 + (0.980 + 0.195i)T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
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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

−22.49000517320084474876657921067, −21.54241716186690644442410984694, −21.04447215534112615624400010410, −20.062925938560214363501534977094, −19.17791823878260169922322301722, −18.356822240322302664900605649292, −17.3036048345822543478230537250, −16.749275911741221577842853813, −16.12143382139928974677484304285, −14.76856464016587510189912960468, −14.48640611223056716879022275740, −13.74144281684688423959769848427, −12.330943909199853604692404627040, −11.41282012270064696184492388801, −10.75870627050547934431833207829, −10.055900023678518673674860352174, −9.40492983186005859587913160358, −8.06572004110935136676818287446, −7.171955279595346415849928635333, −6.19211122622739230681888073179, −5.399646822689474561851263838191, −4.18472737449446647638873840831, −3.53807753914446398831323608920, −2.51503842064801928007878995615, −0.77042885045556561048721722092, 1.051086478620761307402021806737, 1.94108658514431576836090418847, 2.91482701432983201639099190926, 4.69773146939534384030303798535, 5.24526250064011114380091966980, 6.08107384530704127066151907283, 7.2634935345298997441666538381, 7.89144302316847578672965292960, 9.04254588605436488803876690379, 9.54453830549752314421324739373, 10.94407750365958569563358537221, 11.990747952070109626893763130436, 12.41112318143396559845272585586, 12.96290512470179517188301852700, 14.11214616193910583315763922057, 14.91435380091844722088025451667, 15.92228933351792230293993349864, 16.89172962195997701614769251110, 17.54694282480571355798891186558, 18.06880540637791600671505260874, 19.21301955034288904229222496688, 19.736933325242700152293310971070, 20.60709246300627697822649095903, 21.69255226140632888768319213837, 22.201027459864299401916718008867

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