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Properties

Label 2-2385-265.74-c0-0-0
Degree $2$
Conductor $2385$
Sign $0.129 - 0.991i$
Analytic cond. $1.19027$
Root an. cond. $1.09099$
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  + (−1.78 − 0.556i)2-s + (2.05 + 1.41i)4-s + (0.0603 − 0.998i)5-s + (−1.72 − 2.20i)8-s + (−0.663 + 1.74i)10-s + (0.970 + 2.56i)16-s + (0.219 + 1.81i)17-s + (−1.68 − 0.308i)19-s + (1.53 − 1.96i)20-s + (−1.37 + 1.37i)23-s + (−0.992 − 0.120i)25-s + (−0.987 + 1.63i)31-s + (−0.140 − 2.31i)32-s + (0.614 − 3.35i)34-s + (2.83 + 1.48i)38-s + ⋯
L(s)  = 1  + (−1.78 − 0.556i)2-s + (2.05 + 1.41i)4-s + (0.0603 − 0.998i)5-s + (−1.72 − 2.20i)8-s + (−0.663 + 1.74i)10-s + (0.970 + 2.56i)16-s + (0.219 + 1.81i)17-s + (−1.68 − 0.308i)19-s + (1.53 − 1.96i)20-s + (−1.37 + 1.37i)23-s + (−0.992 − 0.120i)25-s + (−0.987 + 1.63i)31-s + (−0.140 − 2.31i)32-s + (0.614 − 3.35i)34-s + (2.83 + 1.48i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (604, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2055162721\)
\(L(\frac12)\) \(\approx\) \(0.2055162721\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.0603 + 0.998i)T \)
53 \( 1 + (0.616 - 0.787i)T \)
good2 \( 1 + (1.78 + 0.556i)T + (0.822 + 0.568i)T^{2} \)
7 \( 1 + (0.568 - 0.822i)T^{2} \)
11 \( 1 + (-0.885 - 0.464i)T^{2} \)
13 \( 1 + (0.354 - 0.935i)T^{2} \)
17 \( 1 + (-0.219 - 1.81i)T + (-0.970 + 0.239i)T^{2} \)
19 \( 1 + (1.68 + 0.308i)T + (0.935 + 0.354i)T^{2} \)
23 \( 1 + (1.37 - 1.37i)T - iT^{2} \)
29 \( 1 + (-0.885 + 0.464i)T^{2} \)
31 \( 1 + (0.987 - 1.63i)T + (-0.464 - 0.885i)T^{2} \)
37 \( 1 + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (0.464 - 0.885i)T^{2} \)
43 \( 1 + (-0.748 - 0.663i)T^{2} \)
47 \( 1 + (-0.239 + 0.269i)T + (-0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.120 - 0.992i)T^{2} \)
61 \( 1 + (-0.970 + 0.760i)T + (0.239 - 0.970i)T^{2} \)
67 \( 1 + (-0.935 + 0.354i)T^{2} \)
71 \( 1 + (0.663 - 0.748i)T^{2} \)
73 \( 1 + (-0.239 - 0.970i)T^{2} \)
79 \( 1 + (-0.115 + 0.0359i)T + (0.822 - 0.568i)T^{2} \)
83 \( 1 + (1.16 + 1.16i)T + iT^{2} \)
89 \( 1 + (-0.970 + 0.239i)T^{2} \)
97 \( 1 + (-0.120 + 0.992i)T^{2} \)
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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

−9.273029015241460765326053100967, −8.524243151628716162490632797233, −8.250208583872593923696388521630, −7.43769744098338984922707761659, −6.43576954654920730912606816473, −5.68917092172578848183781194472, −4.27502985016607058597398762866, −3.42753698681277905194400625105, −1.96857504492954019451413168266, −1.50186613542306795558821365454, 0.23502453722026860380304678044, 2.04824447786473352903454634539, 2.59926199746871322793953336802, 4.07437749166031597566424699881, 5.50803839378407077116692890077, 6.35660492035544300983435072056, 6.78479906801186691114654375482, 7.60982921793906252795593484568, 8.151097723784066724647983399181, 8.948538778457495403352258606782

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