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Properties

Label 2-3420-15.8-c1-0-3
Degree $2$
Conductor $3420$
Sign $-0.993 - 0.117i$
Analytic cond. $27.3088$
Root an. cond. $5.22578$
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  + (−2.20 − 0.398i)5-s + (−2.63 + 2.63i)7-s + 5.14i·11-s + (5.01 + 5.01i)13-s + (−1.61 − 1.61i)17-s + i·19-s + (6.03 − 6.03i)23-s + (4.68 + 1.75i)25-s + 2.85·29-s − 4.48·31-s + (6.85 − 4.75i)35-s + (−4.69 + 4.69i)37-s + 5.94i·41-s + (2.59 + 2.59i)43-s + (−1.63 − 1.63i)47-s + ⋯
L(s)  = 1  + (−0.983 − 0.178i)5-s + (−0.997 + 0.997i)7-s + 1.55i·11-s + (1.39 + 1.39i)13-s + (−0.392 − 0.392i)17-s + 0.229i·19-s + (1.25 − 1.25i)23-s + (0.936 + 0.351i)25-s + 0.530·29-s − 0.805·31-s + (1.15 − 0.803i)35-s + (−0.772 + 0.772i)37-s + 0.928i·41-s + (0.395 + 0.395i)43-s + (−0.237 − 0.237i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.993 - 0.117i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7556430029\)
\(L(\frac12)\) \(\approx\) \(0.7556430029\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.20 + 0.398i)T \)
19 \( 1 - iT \)
good7 \( 1 + (2.63 - 2.63i)T - 7iT^{2} \)
11 \( 1 - 5.14iT - 11T^{2} \)
13 \( 1 + (-5.01 - 5.01i)T + 13iT^{2} \)
17 \( 1 + (1.61 + 1.61i)T + 17iT^{2} \)
23 \( 1 + (-6.03 + 6.03i)T - 23iT^{2} \)
29 \( 1 - 2.85T + 29T^{2} \)
31 \( 1 + 4.48T + 31T^{2} \)
37 \( 1 + (4.69 - 4.69i)T - 37iT^{2} \)
41 \( 1 - 5.94iT - 41T^{2} \)
43 \( 1 + (-2.59 - 2.59i)T + 43iT^{2} \)
47 \( 1 + (1.63 + 1.63i)T + 47iT^{2} \)
53 \( 1 + (8.76 - 8.76i)T - 53iT^{2} \)
59 \( 1 + 0.536T + 59T^{2} \)
61 \( 1 - 4.94T + 61T^{2} \)
67 \( 1 + (5.49 - 5.49i)T - 67iT^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 + (5.75 + 5.75i)T + 73iT^{2} \)
79 \( 1 + 5.62iT - 79T^{2} \)
83 \( 1 + (-9.61 + 9.61i)T - 83iT^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−9.150521045539428572297557257555, −8.361024460016520738568468596357, −7.37065733033740374778670701762, −6.64583374501312547477676155599, −6.26475018536577450718121502854, −4.89776386577854896636688807932, −4.43780362683785628106442400603, −3.47300497235343035564882725398, −2.61388774850668973579168034375, −1.45736762297527978591783174032, 0.27709496346977715227166745945, 1.06648333565900862277328578067, 3.11525767679381964538679352860, 3.42628960765838111525414409911, 3.99603324881770188186849437681, 5.35045650120185602428303885315, 6.00024588984971827762681598892, 6.88997030205701721408114945490, 7.40993966201050290815228429663, 8.419372071374598835105461678223

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