↑

Properties

Label 2-755-5.4-c1-0-67
Degree $2$
Conductor $755$
Sign $-0.579 - 0.814i$
Analytic cond. $6.02870$
Root an. cond. $2.45534$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.95i·2-s − 2.37i·3-s − 1.82·4-s + (1.29 + 1.82i)5-s − 4.64·6-s − 0.105i·7-s − 0.336i·8-s − 2.63·9-s + (3.56 − 2.53i)10-s − 4.15·11-s + 4.34i·12-s − 5.32i·13-s − 0.206·14-s + (4.32 − 3.07i)15-s − 4.31·16-s + 0.871i·17-s + ⋯
L(s)  = 1  − 1.38i·2-s − 1.37i·3-s − 0.913·4-s + (0.579 + 0.814i)5-s − 1.89·6-s − 0.0398i·7-s − 0.119i·8-s − 0.879·9-s + (1.12 − 0.802i)10-s − 1.25·11-s + 1.25i·12-s − 1.47i·13-s − 0.0551·14-s + (1.11 − 0.794i)15-s − 1.07·16-s + 0.211i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(755\)    =    \(5 \cdot 151\)
Sign: $-0.579 - 0.814i$
Analytic conductor: \(6.02870\)
Root analytic conductor: \(2.45534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{755} (454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 755,\ (\ :1/2),\ -0.579 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566724 + 1.09893i\)
\(L(\frac12)\) \(\approx\) \(0.566724 + 1.09893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.29 - 1.82i)T \)
151 \( 1 + T \)
good2 \( 1 + 1.95iT - 2T^{2} \)
3 \( 1 + 2.37iT - 3T^{2} \)
7 \( 1 + 0.105iT - 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 + 5.32iT - 13T^{2} \)
17 \( 1 - 0.871iT - 17T^{2} \)
19 \( 1 + 3.02T + 19T^{2} \)
23 \( 1 + 0.483iT - 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
31 \( 1 - 0.851T + 31T^{2} \)
37 \( 1 + 9.23iT - 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 0.931iT - 43T^{2} \)
47 \( 1 + 0.990iT - 47T^{2} \)
53 \( 1 + 3.79iT - 53T^{2} \)
59 \( 1 - 1.29T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 + 4.44iT - 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 5.76iT - 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 + 13.9iT - 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 6.83iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25671375554823814255264939529, −9.156517587456597706124446323739, −7.87194802469393762805782140631, −7.35083519025632420147347925437, −6.25733012474180329830175475144, −5.42082870985740952805851176673, −3.68343526484261490715178696269, −2.54626743476659594319311866975, −2.12138023327038826280705533675, −0.59209506903360400621265119370, 2.30344571475724335479193053625, 4.13076391829023292831830808739, 4.81549028704613558768175132585, 5.48283156288264770024786769913, 6.33869695206672308289244071647, 7.47302252010540429260970160666, 8.429122148026983952915657287651, 9.153768137315088015664788482040, 9.680706959171208510698098855588, 10.68760554063911794556980808905

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