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Properties

Label 2-3420-95.18-c1-0-8
Degree $2$
Conductor $3420$
Sign $-0.966 - 0.256i$
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  + (1.48 + 1.67i)5-s + (−1.48 + 1.48i)7-s − 0.806·11-s + (−0.437 + 0.437i)13-s + (−3.15 + 3.15i)17-s + (1.81 − 3.96i)19-s + (−1.86 − 1.86i)23-s + (−0.612 + 4.96i)25-s + 4.50·29-s + 6.67i·31-s + (−4.67 − 0.287i)35-s + (5.29 + 5.29i)37-s − 11.1i·41-s + (−3.86 − 3.86i)43-s + (−6.83 + 6.83i)47-s + ⋯
L(s)  = 1  + (0.662 + 0.749i)5-s + (−0.559 + 0.559i)7-s − 0.243·11-s + (−0.121 + 0.121i)13-s + (−0.765 + 0.765i)17-s + (0.416 − 0.909i)19-s + (−0.389 − 0.389i)23-s + (−0.122 + 0.992i)25-s + 0.836·29-s + 1.19i·31-s + (−0.790 − 0.0485i)35-s + (0.870 + 0.870i)37-s − 1.74i·41-s + (−0.590 − 0.590i)43-s + (−0.996 + 0.996i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8741864695\)
\(L(\frac12)\) \(\approx\) \(0.8741864695\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.67i)T \)
19 \( 1 + (-1.81 + 3.96i)T \)
good7 \( 1 + (1.48 - 1.48i)T - 7iT^{2} \)
11 \( 1 + 0.806T + 11T^{2} \)
13 \( 1 + (0.437 - 0.437i)T - 13iT^{2} \)
17 \( 1 + (3.15 - 3.15i)T - 17iT^{2} \)
23 \( 1 + (1.86 + 1.86i)T + 23iT^{2} \)
29 \( 1 - 4.50T + 29T^{2} \)
31 \( 1 - 6.67iT - 31T^{2} \)
37 \( 1 + (-5.29 - 5.29i)T + 37iT^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + (3.86 + 3.86i)T + 43iT^{2} \)
47 \( 1 + (6.83 - 6.83i)T - 47iT^{2} \)
53 \( 1 + (8.92 - 8.92i)T - 53iT^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + (-4.94 - 4.94i)T + 67iT^{2} \)
71 \( 1 - 3.04iT - 71T^{2} \)
73 \( 1 + (6.19 + 6.19i)T + 73iT^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 + (5.32 + 5.32i)T + 83iT^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (-7.46 - 7.46i)T + 97iT^{2} \)
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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

−9.099900413255336767839041573871, −8.262777363491253604804396524198, −7.33188614220928184771767218073, −6.50340324066801904960456275713, −6.19423603688733104650101758722, −5.21883798732226649028049204675, −4.38496708079869519927394481922, −3.13453300557219947100729955535, −2.65596120214440589531009368807, −1.59466523168567976037905965394, 0.24845316158913749991315940818, 1.46802387346851780706669926522, 2.53344520886463749148672834685, 3.53861516065485341057492119675, 4.52409672494745959521250020304, 5.13760398043681499478118140476, 6.12119860783670996750508007306, 6.57312827289069659001572218159, 7.69314870588902836642199743051, 8.153664667050577105269681855369

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