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Properties

Label 2-3040-760.189-c0-0-5
Degree $2$
Conductor $3040$
Sign $0.923 + 0.382i$
Analytic cond. $1.51715$
Root an. cond. $1.23172$
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  + 1.84i·3-s − i·5-s − 2.41·9-s − 1.41i·11-s − 0.765i·13-s + 1.84·15-s − i·19-s − 25-s − 2.61i·27-s + 2.61·33-s − 1.84i·37-s + 1.41·39-s + 2.41i·45-s + 49-s + 1.84i·53-s + ⋯
L(s)  = 1  + 1.84i·3-s − i·5-s − 2.41·9-s − 1.41i·11-s − 0.765i·13-s + 1.84·15-s − i·19-s − 25-s − 2.61i·27-s + 2.61·33-s − 1.84i·37-s + 1.41·39-s + 2.41i·45-s + 49-s + 1.84i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9743055292\)
\(L(\frac12)\) \(\approx\) \(0.9743055292\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 - 1.84iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + 0.765iT - T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.84iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.84iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + 0.765iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 0.765T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.131770621772104419048712972980, −8.420207062584904724402048665471, −7.68179270341263187089234390987, −6.15610170172252544845102570537, −5.52696992948438816956856046675, −5.01101022141876503627117339849, −4.15597820444733265786945905408, −3.50308515196843513419202503088, −2.61170588810849959896556865801, −0.59951078393728761524774019815, 1.52151892859822539392256736196, 2.14794587656025783458366885769, 3.00581128274858847579974132453, 4.16058687656121827984753183799, 5.40394562837852720446623244721, 6.28981292091680665656525615313, 6.82140373791270475307627008612, 7.28205544030960554885421401394, 7.937091134110833100646298850130, 8.662030668196823694125657146319

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