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Properties

Label 2-3040-760.189-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9803430572\)
\(L(\frac12)\) \(\approx\) \(0.9803430572\)
\(L(1)\) 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

−9.538659619798452159825103328905, −8.830648344599126519386748547400, −7.81018848235131387722119106941, −7.19073717626386772539950433424, −5.97521615628846796822376442194, −5.50546884827146270707507362751, −4.46125972973060929605443694584, −3.92966766740066944821535809748, −3.09963862114343230404873090610, −2.23193219135662575691594295956, 0.60574365584322645523386566179, 1.50130767339970013394648671788, 2.48451050333165616733137667715, 3.49814197386740149013981179238, 4.81570471791419315592785597188, 5.60341480886680549451571103805, 6.36696421990738248444175953234, 6.89577294499925082311403247035, 7.81426585951334064678575930535, 8.478139252639691379492295594409

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