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Properties

Label 2-3040-3040.1709-c0-0-4
Degree $2$
Conductor $3040$
Sign $0.0980 + 0.995i$
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  + (−0.995 − 0.0980i)2-s + (0.591 − 1.42i)3-s + (0.980 + 0.195i)4-s + (0.923 − 0.382i)5-s + (−0.728 + 1.36i)6-s + (−0.956 − 0.290i)8-s + (−0.983 − 0.983i)9-s + (−0.956 + 0.290i)10-s + (0.636 + 1.53i)11-s + (0.858 − 1.28i)12-s + (0.871 + 0.360i)13-s − 1.54i·15-s + (0.923 + 0.382i)16-s + (0.881 + 1.07i)18-s + (−0.923 − 0.382i)19-s + (0.980 − 0.195i)20-s + ⋯
L(s)  = 1  + (−0.995 − 0.0980i)2-s + (0.591 − 1.42i)3-s + (0.980 + 0.195i)4-s + (0.923 − 0.382i)5-s + (−0.728 + 1.36i)6-s + (−0.956 − 0.290i)8-s + (−0.983 − 0.983i)9-s + (−0.956 + 0.290i)10-s + (0.636 + 1.53i)11-s + (0.858 − 1.28i)12-s + (0.871 + 0.360i)13-s − 1.54i·15-s + (0.923 + 0.382i)16-s + (0.881 + 1.07i)18-s + (−0.923 − 0.382i)19-s + (0.980 − 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $0.0980 + 0.995i$
Analytic conductor: \(1.51715\)
Root analytic conductor: \(1.23172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3040} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :0),\ 0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245737460\)
\(L(\frac12)\) \(\approx\) \(1.245737460\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 + 0.0980i)T \)
5 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
good3 \( 1 + (-0.591 + 1.42i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.871 - 0.360i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.181 - 0.0750i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.485 + 1.17i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.425 + 1.02i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - 1.76T + T^{2} \)
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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

−8.716289825492438216496791974558, −8.084157944495914997275874457612, −7.26221066325778350029732035729, −6.47230132403868526080424728566, −6.41918880119754680266884080893, −5.00430478351938467456721270915, −3.69734718665689455371845026995, −2.34094292759677663204172320511, −1.94392415194642290846764304539, −1.15999895981725472859911522484, 1.37970672401797491502599746941, 2.70922300597204621083823141436, 3.30679742886735908734097561923, 4.18133087434296641415850877070, 5.56427844157965968615865063641, 5.99199530045662340610280946238, 6.75751368702626672892865887383, 7.985086920269625525696497587609, 8.731882165194199659065147504137, 8.945054352634447944150825155613

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