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Properties

Label 2-2380-85.84-c1-0-12
Degree $2$
Conductor $2380$
Sign $-0.578 - 0.815i$
Analytic cond. $19.0043$
Root an. cond. $4.35940$
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  + 1.60·3-s + (−0.593 + 2.15i)5-s + 7-s − 0.409·9-s + 1.11i·11-s − 0.620i·13-s + (−0.955 + 3.47i)15-s + (−2.60 + 3.19i)17-s + 2.31·19-s + 1.60·21-s − 5.06·23-s + (−4.29 − 2.55i)25-s − 5.48·27-s + 8.48i·29-s + 7.02i·31-s + ⋯
L(s)  = 1  + 0.929·3-s + (−0.265 + 0.964i)5-s + 0.377·7-s − 0.136·9-s + 0.337i·11-s − 0.172i·13-s + (−0.246 + 0.895i)15-s + (−0.632 + 0.774i)17-s + 0.530·19-s + 0.351·21-s − 1.05·23-s + (−0.859 − 0.511i)25-s − 1.05·27-s + 1.57i·29-s + 1.26i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.578 - 0.815i$
Analytic conductor: \(19.0043\)
Root analytic conductor: \(4.35940\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :1/2),\ -0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.634523982\)
\(L(\frac12)\) \(\approx\) \(1.634523982\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.593 - 2.15i)T \)
7 \( 1 - T \)
17 \( 1 + (2.60 - 3.19i)T \)
good3 \( 1 - 1.60T + 3T^{2} \)
11 \( 1 - 1.11iT - 11T^{2} \)
13 \( 1 + 0.620iT - 13T^{2} \)
19 \( 1 - 2.31T + 19T^{2} \)
23 \( 1 + 5.06T + 23T^{2} \)
29 \( 1 - 8.48iT - 29T^{2} \)
31 \( 1 - 7.02iT - 31T^{2} \)
37 \( 1 - 5.36T + 37T^{2} \)
41 \( 1 + 3.80iT - 41T^{2} \)
43 \( 1 - 1.63iT - 43T^{2} \)
47 \( 1 - 0.311iT - 47T^{2} \)
53 \( 1 + 7.23iT - 53T^{2} \)
59 \( 1 + 7.16T + 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 - 8.09iT - 67T^{2} \)
71 \( 1 + 6.35iT - 71T^{2} \)
73 \( 1 - 9.07T + 73T^{2} \)
79 \( 1 + 6.96iT - 79T^{2} \)
83 \( 1 - 8.78iT - 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−9.075013802500636701130490165728, −8.428825445271719641477246199136, −7.77309031003751208770165285459, −7.09479009330013290417267053707, −6.26624233990063617629214018754, −5.31923781920615110455364685800, −4.17492779818593791848462767737, −3.41679515411724482920488564384, −2.63567409213374621078168544632, −1.70992973998060481203100276542, 0.45481199400094583191186051555, 1.90171078482946323171056928280, 2.77081914877863778537497492238, 3.91542750174290701417157371901, 4.51546465111722515806762592889, 5.52919367207064474071405908553, 6.28210921106110307170286519414, 7.66057800574994109048298589808, 7.936651902008088742441722288310, 8.633274188822789993089333416395

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