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Properties

Label 1-760-760.307-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.949 + 0.313i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 0.766i)3-s + (0.866 + 0.5i)7-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.984 + 0.173i)17-s + (0.173 + 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.866 + 0.5i)27-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (0.342 − 0.939i)33-s − i·37-s + 39-s + (−0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.866 + 0.5i)7-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.984 + 0.173i)17-s + (0.173 + 0.984i)21-s + (−0.342 − 0.939i)23-s + (−0.866 + 0.5i)27-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s + (0.342 − 0.939i)33-s − i·37-s + 39-s + (−0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2445691508 + 1.523055175i\)
\(L(\frac12)\) \(\approx\) \(0.2445691508 + 1.523055175i\)
\(L(1)\) \(\approx\) \(1.117162518 + 0.4681846828i\)
\(L(1)\) \(\approx\) \(1.117162518 + 0.4681846828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.642 - 0.766i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.642 - 0.766i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−21.69266341988380691503666934879, −20.68793004306569139928959915533, −20.426887345037161084323388725302, −19.42773311896267974817879811033, −18.61682081645836169422020922775, −17.772058672925977482252442394819, −17.38261067286890057587211717889, −16.037252705274256907346233956223, −15.144694516959840957147945136543, −14.433482663510439513663988987340, −13.50164158411262808441933694348, −13.1280811625554774575600869069, −11.83919169157858481618853519852, −11.31060586310392147955526929589, −10.10338639559708241795827957474, −9.13470281993617862367714313893, −8.30181627459854928488665546848, −7.45361038308710701043360739839, −6.86678259838434411752877023050, −5.68230783535816550614737943952, −4.445302960117796337543262613577, −3.667491192821603342549067419406, −2.15035365565159932632578277056, −1.73634765794197865684927423049, −0.28731868279981875197038002775, 1.44627506461586720620060046611, 2.63337753331083960124332126652, 3.40144643434368221358088772088, 4.59369829341156472475378758247, 5.27581952805658752662599998727, 6.311484553821096590817471022454, 7.75954330983725400750467875540, 8.53245749390288167587251815568, 8.87078950629118737230482413466, 10.2800047890931447800018181107, 10.81736300231293967841590340476, 11.61115664514469167863522476003, 12.935632189778778656095883587920, 13.632802950043949988506274558153, 14.58151464418643215087104567480, 15.17328123648315863555731023226, 15.99242590294608672201899721664, 16.653938415119455683010291677471, 17.95078258899841187464979278589, 18.40983614097568743438117461433, 19.52254497789255605322028571207, 20.3355086876113374184936667860, 20.90751284391703665355269067046, 21.77486436061778322381074862112, 22.20278145678105860868488450173

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