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Properties

Label 2-2940-35.4-c1-0-3
Degree $2$
Conductor $2940$
Sign $-0.652 - 0.758i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.133 − 2.23i)5-s + (0.499 + 0.866i)9-s + (−2 + 3.46i)11-s + 2i·13-s + (1 − 1.99i)15-s + (1.73 + i)17-s + (−1 − 1.73i)19-s + (−5.19 + 3i)23-s + (−4.96 + 0.598i)25-s + 0.999i·27-s − 6·29-s + (−3 + 5.19i)31-s + (−3.46 + 1.99i)33-s + (3.46 − 2i)37-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.0599 − 0.998i)5-s + (0.166 + 0.288i)9-s + (−0.603 + 1.04i)11-s + 0.554i·13-s + (0.258 − 0.516i)15-s + (0.420 + 0.242i)17-s + (−0.229 − 0.397i)19-s + (−1.08 + 0.625i)23-s + (−0.992 + 0.119i)25-s + 0.192i·27-s − 1.11·29-s + (−0.538 + 0.933i)31-s + (−0.603 + 0.348i)33-s + (0.569 − 0.328i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.652 - 0.758i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.652 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8881714720\)
\(L(\frac12)\) \(\approx\) \(0.8881714720\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.46 + 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (3.46 - 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.73 - i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (-8 - 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.206042313400886797760568069374, −8.227998072614270052990858249509, −7.70973580323443419863822601132, −6.94384500647124698800297391187, −5.76547911886303417729779599806, −5.08295774549015950604946799226, −4.30850671565078949803722917557, −3.63416962705288610782299724849, −2.30600488266309567015034076772, −1.53500201938830136715055678266, 0.24171446108791185581727801293, 1.88350556712224397429343817048, 2.85520428056331316398722553554, 3.42338309065850821338364508709, 4.37426460407881051287396550506, 5.88050412939606147294929585546, 5.93626455951061175937823400600, 7.17214924081788713089749111376, 7.75195061424548306038158704563, 8.280012875117982510488016914379

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