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Properties

Label 2-2380-2380.1903-c0-0-8
Degree $2$
Conductor $2380$
Sign $0.973 - 0.229i$
Analytic cond. $1.18777$
Root an. cond. $1.08985$
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  − 2-s + 4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − 8-s + i·9-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 0.707i)17-s − i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + (0.707 + 0.707i)28-s + 1.41i·31-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − 8-s + i·9-s + (−0.707 + 0.707i)10-s + (−0.707 − 0.707i)14-s + 16-s + (0.707 − 0.707i)17-s − i·18-s + (0.707 − 0.707i)20-s − 1.00i·25-s + (0.707 + 0.707i)28-s + 1.41i·31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(1.18777\)
Root analytic conductor: \(1.08985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (1903, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9823075270\)
\(L(\frac12)\) \(\approx\) \(0.9823075270\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.197254291317784498264669638273, −8.357092623628522631016084520290, −8.002268701920795571169499402705, −7.07261138901589739445447813572, −6.05573965643822305986809359963, −5.31451161869299483610420932335, −4.73661629593216491365939132681, −3.02505594701188596318640213934, −2.12056570304014836819502942223, −1.31643084810390214983089311770, 1.09321258104732669019538978730, 2.10886590224226793747273203715, 3.21140544484599916813339562472, 4.09194288371541402748277665214, 5.58468619612288767465988911583, 6.17223378685330214154346784410, 6.98314755674405347704195742191, 7.59546343387826837617772859103, 8.371103015653168086450178194334, 9.251478988376680099352666673992

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