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Properties

Label 1-764-764.739-r0-0-0
Degree $1$
Conductor $764$
Sign $0.791 - 0.611i$
Analytic cond. $3.54800$
Root an. cond. $3.54800$
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  + (−0.945 − 0.324i)3-s + (−0.401 + 0.915i)5-s + 7-s + (0.789 + 0.614i)9-s + (−0.0825 − 0.996i)11-s + (0.945 − 0.324i)13-s + (0.677 − 0.735i)15-s + (−0.879 − 0.475i)17-s + (−0.986 − 0.164i)19-s + (−0.945 − 0.324i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.546 − 0.837i)27-s + (0.677 − 0.735i)29-s + (0.789 + 0.614i)31-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)3-s + (−0.401 + 0.915i)5-s + 7-s + (0.789 + 0.614i)9-s + (−0.0825 − 0.996i)11-s + (0.945 − 0.324i)13-s + (0.677 − 0.735i)15-s + (−0.879 − 0.475i)17-s + (−0.986 − 0.164i)19-s + (−0.945 − 0.324i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.546 − 0.837i)27-s + (0.677 − 0.735i)29-s + (0.789 + 0.614i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.791 - 0.611i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9690411034 - 0.3309171316i\)
\(L(\frac12)\) \(\approx\) \(0.9690411034 - 0.3309171316i\)
\(L(1)\) \(\approx\) \(0.8486689990 - 0.07533049296i\)
\(L(1)\) \(\approx\) \(0.8486689990 - 0.07533049296i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
191 \( 1 \)
good3 \( 1 + (-0.945 - 0.324i)T \)
5 \( 1 + (-0.401 + 0.915i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.0825 - 0.996i)T \)
13 \( 1 + (0.945 - 0.324i)T \)
17 \( 1 + (-0.879 - 0.475i)T \)
19 \( 1 + (-0.986 - 0.164i)T \)
23 \( 1 + (0.986 + 0.164i)T \)
29 \( 1 + (0.677 - 0.735i)T \)
31 \( 1 + (0.789 + 0.614i)T \)
37 \( 1 + (-0.789 + 0.614i)T \)
41 \( 1 + (-0.546 + 0.837i)T \)
43 \( 1 + (-0.245 - 0.969i)T \)
47 \( 1 + (-0.0825 - 0.996i)T \)
53 \( 1 + (0.0825 + 0.996i)T \)
59 \( 1 + (-0.789 - 0.614i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (0.879 - 0.475i)T \)
71 \( 1 + (0.546 - 0.837i)T \)
73 \( 1 + (0.0825 - 0.996i)T \)
79 \( 1 + (0.677 + 0.735i)T \)
83 \( 1 + (-0.986 + 0.164i)T \)
89 \( 1 + (-0.245 + 0.969i)T \)
97 \( 1 + (0.789 - 0.614i)T \)
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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

−22.69677981581873390355694495982, −21.42449995937680842676633031631, −21.01257445473210547145988340484, −20.33149439645250599330293230142, −19.24269639753112350582559027914, −18.224933140416351984378084483171, −17.386963717853270248577873921863, −17.05446269722801951128915412173, −15.91260081041766337190234913222, −15.41255950603265361911652494596, −14.50944537812818127215691894801, −13.1290793785701647656821959999, −12.58241946847332956929650334189, −11.64549583313847026836634536897, −11.03742244562069779518096498485, −10.2028360218666837027004117597, −8.971664363949528401107001718686, −8.39460850807469372249000687211, −7.20161135319296061347720517050, −6.28991894570824947791994270078, −5.14974314710649875491670706087, −4.54811534638961548359993425048, −3.91962001026606491341074522406, −1.95991956353872053471522619196, −1.05717154618244573365176344358, 0.67929718939891020581386167450, 1.97128542254556180344233258023, 3.18580834693742970515514976837, 4.3674430306765566178427298233, 5.24573760645437519096472406853, 6.34968577590130327689171683272, 6.848116706694745689164963911294, 8.027699319924459334305214760844, 8.62340504508242410026251208303, 10.280675212658319150961426930702, 11.000181298704899140498185284710, 11.301818254603695428474554929782, 12.18870155751320136516685496555, 13.47397633951846162368998508504, 13.89840498605837552894652313872, 15.29930967667172974589758772051, 15.60196017276628993748809365370, 16.83835180436871739954348241520, 17.52561055412764483227229351673, 18.347753097687699224987133828138, 18.79665095125414315387431043378, 19.69782742514322006548274414952, 20.98675648679139544355706845536, 21.58401276150229764860929131063, 22.37062644399267513884451945785

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