↑

Properties

Label 1-760-760.107-r1-0-0
Degree $1$
Conductor $760$
Sign $0.830 - 0.557i$
Analytic cond. $81.6733$
Root an. cond. $81.6733$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s + 31-s + (−0.866 − 0.5i)33-s − i·37-s + 39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s + 31-s + (−0.866 − 0.5i)33-s − i·37-s + 39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.830 - 0.557i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ 0.830 - 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348415143 - 0.4108541780i\)
\(L(\frac12)\) \(\approx\) \(1.348415143 - 0.4108541780i\)
\(L(1)\) \(\approx\) \(0.8403407540 - 0.1795826740i\)
\(L(1)\) \(\approx\) \(0.8403407540 - 0.1795826740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.38746250386227883001964678208, −21.5633231209209387264572805492, −21.00800735179494043012435068783, −19.70801225069039558229093358373, −19.171167732082352374933565382705, −17.94124725577537588341261497441, −17.46183567337460818530237453193, −16.730438748833105677865584021234, −15.62014993089607939938879274229, −15.2121295387929510424221434518, −14.33323157789375834846602804171, −12.97080203551161172242757645227, −12.25845522317852121569636101475, −11.55851937351043668438573021452, −10.80577582250209569141076476420, −9.66492435041833643676598523797, −9.18579111910195944509812989605, −8.04505059455655080912173264768, −6.73148123506282569460554216593, −6.1134751916098536522092350122, −5.13877062260936629136676809192, −4.38638088063855151645062638984, −3.2237680640295592307541279863, −1.99579250558704846187771722416, −0.61156463401315822224093876260, 0.65102383948248942483058249623, 1.51406309231364768324735897852, 2.837539172604692865437240930513, 4.38856619044779753400326122242, 4.74782146463877940119908832011, 6.19045349221030703135978415395, 6.89418075253377655991149422609, 7.42532770147429126973516818892, 8.72677056213699097900719859876, 9.761807815996288031432152579893, 10.66109287793425474726902116774, 11.419790698862953501646291489102, 12.154698943495192074202041160277, 13.04609436332407561623697705439, 13.885609837072978426257384229716, 14.600161768135263207963615182021, 15.88880786343774037312420226036, 16.67596670132977874534330312481, 17.286953652567164537441883372967, 17.83404013997740064903947701961, 19.04462537252965248388087795975, 19.53848387063262470381515328407, 20.44168056517641336851435097028, 21.48814181394840085823920971411, 22.4509051300564475910468509576

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