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Properties

Label 2-2380-2380.1427-c0-0-18
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  − i·2-s − 4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s + i·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 − 18-s + (0.707 + 0.707i)20-s + 1.00i·25-s + (−0.707 + 0.707i)28-s − 1.41i·31-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)7-s + i·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 − 18-s + (0.707 + 0.707i)20-s + 1.00i·25-s + (−0.707 + 0.707i)28-s − 1.41i·31-s − i·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: $\chi_{2380} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7839367746\)
\(L(\frac12)\) \(\approx\) \(0.7839367746\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
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

−8.996609065502955061692792304982, −8.060195084873142323700984205922, −7.53357421066440524075847683160, −6.38764215521442081737001694948, −5.22104919491739287854534410689, −4.45854022370546569958032718972, −3.95796379500301565440331101696, −2.98417375564784335914781607416, −1.61762039679279264893636010816, −0.55344993039045456239274174995, 1.87926495496926783563178486210, 3.12738448427431443570941624584, 4.20826606834830570860546507677, 4.92909894037253855737162102802, 5.67238911821466429024931359381, 6.64131933019315788044050226773, 7.23835291840296568325688026906, 8.192195976582885651316398266662, 8.350109364106543044480488958869, 9.273522354957300804769774313860

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