Properties

Label 1-760-760.637-r0-0-0
Degree $1$
Conductor $760$
Sign $0.391 - 0.920i$
Analytic cond. $3.52942$
Root an. cond. $3.52942$
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  + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s i·37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (0.866 − 0.5i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.342 − 0.939i)17-s + (−0.939 + 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.866 − 0.5i)27-s + (0.939 + 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.642 + 0.766i)33-s i·37-s − 39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ 0.391 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.002674537 - 0.6630913597i\)
\(L(\frac12)\) \(\approx\) \(1.002674537 - 0.6630913597i\)
\(L(1)\) \(\approx\) \(0.9188633732 - 0.2324428548i\)
\(L(1)\) \(\approx\) \(0.9188633732 - 0.2324428548i\)

Euler product

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

−22.51192544445938326940694639322, −21.79217123320085150654337729169, −21.112742548737483334864039217200, −20.33985389448912792233393869207, −19.17491793129650829886231431685, −18.348692111447890134819923477999, −17.474496608136799314551665450585, −17.27657777956227689210861062121, −15.9098451753869759454309085938, −15.44817655213533407155733269816, −14.52332811787704395316048886706, −13.476113611074753954385841187713, −12.421270217191723995093402002184, −11.809746805224166748903780661594, −11.06208391906885981021560500933, −10.25877718956784677085176643036, −9.26305525429913558580571350989, −8.30669032225067432425587218708, −7.26874589519781680811913973121, −6.23014086927650299981594324579, −5.60544645611954116397956998303, −4.47867591014733867344901250328, −3.898348286830783427142910446124, −2.09901078507284976056774645352, −1.25836356061658474180637060023, 0.76323480452570374323434214462, 1.614978483449765021582312044313, 3.20241309665892877899728026535, 4.40789815946397385644389286221, 5.06259072178654245104245821156, 6.22110465226219435305838363090, 6.77436435661985115376745700091, 7.979867144718211019319218396756, 8.68528322603829724123038366889, 10.05368643678848556360124483066, 10.80598357332876532407843224593, 11.5022299954272612806849588377, 12.0917425766141294994573268126, 13.39639416517192403712842714622, 13.84321813418207440883780416482, 14.93322257187217957267992729516, 16.177969481093299761162476616214, 16.418070808264133306407273410300, 17.57115781722170469383318428051, 18.05869803833861536497106835165, 18.800977881971282297090837057281, 19.89892499725374780011473213759, 20.7616683614443650116616211796, 21.55428516959674702080103573901, 22.282249028357560056383552914071

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