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Properties

Label 1-764-764.643-r0-0-0
Degree $1$
Conductor $764$
Sign $0.971 + 0.237i$
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  + (−0.245 + 0.969i)3-s + (−0.0825 − 0.996i)5-s + 7-s + (−0.879 − 0.475i)9-s + (0.945 + 0.324i)11-s + (0.245 + 0.969i)13-s + (0.986 + 0.164i)15-s + (−0.401 − 0.915i)17-s + (0.789 − 0.614i)19-s + (−0.245 + 0.969i)21-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.677 − 0.735i)27-s + (0.986 + 0.164i)29-s + (−0.879 − 0.475i)31-s + ⋯
L(s)  = 1  + (−0.245 + 0.969i)3-s + (−0.0825 − 0.996i)5-s + 7-s + (−0.879 − 0.475i)9-s + (0.945 + 0.324i)11-s + (0.245 + 0.969i)13-s + (0.986 + 0.164i)15-s + (−0.401 − 0.915i)17-s + (0.789 − 0.614i)19-s + (−0.245 + 0.969i)21-s + (−0.789 + 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.677 − 0.735i)27-s + (0.986 + 0.164i)29-s + (−0.879 − 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(3.54800\)
Root analytic conductor: \(3.54800\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (0:\ ),\ 0.971 + 0.237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.502285703 + 0.1812416278i\)
\(L(\frac12)\) \(\approx\) \(1.502285703 + 0.1812416278i\)
\(L(1)\) \(\approx\) \(1.127164498 + 0.1388342056i\)
\(L(1)\) \(\approx\) \(1.127164498 + 0.1388342056i\)

Euler product

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

−22.23131853911311997301640447149, −21.98204370232857681955915731439, −20.589023783719539363422855293354, −19.81126438660584944299550774729, −19.07176879719885459808843004309, −18.1056637789032230838772812677, −17.84687143237522625909948250609, −16.977341377828405995792252433568, −15.84418738761428123333204746005, −14.605093544953493748409082308172, −14.378650064320528628512995989897, −13.42936859396949610390821104947, −12.3598278934743250296661577143, −11.62034927185113314921210356630, −10.938573516710008359994825480322, −10.16523954501610959012631721056, −8.61914216738910178834383208957, −7.996732353141335245274062591, −7.18782714132933450958790701037, −6.21731524833756313568541965175, −5.64199504093561083769507512384, −4.20199381166577451175554389696, −3.10566906847242276958596877540, −2.03855323633500900962954913707, −1.06968223731513743840972162766, 0.95745934294030483336774123158, 2.13947985184153995501464291342, 3.73552084687611132422684747186, 4.49383195207182613653067721528, 5.03588506292400004946543029130, 6.05915537606250411399485771131, 7.30468095609434656189069575789, 8.41205609047967494351879800362, 9.25991616574225385601395766305, 9.6045726838929997196895418624, 11.09047524850865180453844677414, 11.59909572452473813893157544798, 12.21497225683175118947589028779, 13.68852499599387160925629177123, 14.23751963986177954814401196525, 15.21858615306416065133651197341, 16.08524230362963990822692183204, 16.59954563252824571596220656958, 17.536444207000819730076360846009, 18.03224772381350153339142892661, 19.5356274190984851980867863451, 20.2787151700116945438101238693, 20.716450376841031117843887606726, 21.7140218103806922645799383898, 22.100765885700336284494417353059

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