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Properties

Label 1-764-764.227-r1-0-0
Degree $1$
Conductor $764$
Sign $0.820 + 0.571i$
Analytic cond. $82.1032$
Root an. cond. $82.1032$
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  + (0.677 − 0.735i)3-s + (0.245 + 0.969i)5-s − 7-s + (−0.0825 − 0.996i)9-s + (−0.546 − 0.837i)11-s + (−0.677 − 0.735i)13-s + (0.879 + 0.475i)15-s + (0.945 + 0.324i)17-s + (0.401 + 0.915i)19-s + (−0.677 + 0.735i)21-s + (0.401 + 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.789 − 0.614i)27-s + (−0.879 − 0.475i)29-s + (0.0825 + 0.996i)31-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)3-s + (0.245 + 0.969i)5-s − 7-s + (−0.0825 − 0.996i)9-s + (−0.546 − 0.837i)11-s + (−0.677 − 0.735i)13-s + (0.879 + 0.475i)15-s + (0.945 + 0.324i)17-s + (0.401 + 0.915i)19-s + (−0.677 + 0.735i)21-s + (0.401 + 0.915i)23-s + (−0.879 + 0.475i)25-s + (−0.789 − 0.614i)27-s + (−0.879 − 0.475i)29-s + (0.0825 + 0.996i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(764\)    =    \(2^{2} \cdot 191\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(82.1032\)
Root analytic conductor: \(82.1032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{764} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 764,\ (1:\ ),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.823496941 + 0.5722158532i\)
\(L(\frac12)\) \(\approx\) \(1.823496941 + 0.5722158532i\)
\(L(1)\) \(\approx\) \(1.170021440 - 0.06407879560i\)
\(L(1)\) \(\approx\) \(1.170021440 - 0.06407879560i\)

Euler product

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

−22.11853706731786330688612814114, −20.981170254085056706736648554626, −20.73795608653960525044312160415, −19.71305545349783042081536991395, −19.25495781461555283053591454671, −18.09245578293867737190733391720, −16.9020414915704212083501833117, −16.407328309776675723114978160083, −15.73800116914474252961042253515, −14.80113260358650739532732927004, −13.98666326344571913281400729600, −12.95995422857693384806923996144, −12.57513468532832271232234430559, −11.3087259762057594644618705009, −10.096316443980927775960790351291, −9.49058546081796799223660651692, −9.088459494200599860443214831803, −7.86305768257556530729672818391, −7.06779415825347362628519464597, −5.66314636704400654489548488514, −4.828512498271817571744495886227, −4.09679110055579148212420058197, −2.87979894133336028834351093262, −2.07593418567661120759547751830, −0.46466078497929674802995827254, 0.89131082855452564642790922616, 2.25480043224405745829278727353, 3.22434343548329279508379875299, 3.49466381551316492734514532004, 5.60888707634058910120994963284, 6.09654213804525051099569455560, 7.30873261187225890091605860250, 7.673276611348755083649952964063, 8.857729311725130068549384646653, 9.89080263628693118612912414890, 10.40275910636426697321593902027, 11.69210015237249172300739575375, 12.55722939795183939980784719710, 13.351089302472936208239935489123, 14.002505228158614734126680738750, 14.84746178504546385598037452360, 15.56087304944682070332788425097, 16.64006114756169521302350082432, 17.63021237171337068197243127543, 18.46995033304397487571738234712, 19.12309489021431960257200637742, 19.497584107035026211128467308617, 20.64997728379915997157557060249, 21.4330264394425406901799443644, 22.3927087833282623905287032725

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