Properties

Label 2-3420-95.94-c0-0-4
Degree $2$
Conductor $3420$
Sign $0.5 + 0.866i$
Analytic cond. $1.70680$
Root an. cond. $1.30644$
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  + (0.5 + 0.866i)5-s − 1.73i·7-s − 11-s − 1.73i·17-s + 19-s + (−0.499 + 0.866i)25-s + (1.49 − 0.866i)35-s − 1.73i·43-s − 1.73i·47-s − 1.99·49-s + (−0.5 − 0.866i)55-s + 61-s + 1.73i·73-s + 1.73i·77-s + (1.49 − 0.866i)85-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s − 1.73i·7-s − 11-s − 1.73i·17-s + 19-s + (−0.499 + 0.866i)25-s + (1.49 − 0.866i)35-s − 1.73i·43-s − 1.73i·47-s − 1.99·49-s + (−0.5 − 0.866i)55-s + 61-s + 1.73i·73-s + 1.73i·77-s + (1.49 − 0.866i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(1.70680\)
Root analytic conductor: \(1.30644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :0),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230521248\)
\(L(\frac12)\) \(\approx\) \(1.230521248\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
good7 \( 1 + 1.73iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.619621629813422481848007233415, −7.50489465557566279899137031451, −7.28555396692391030310139265831, −6.71034339543966271381691718488, −5.51458192167458795804042464448, −4.96399154522782779481482275731, −3.83739183333108917180272934079, −3.13311005417323666772238798720, −2.20634282294789445149447807980, −0.73993602968990485844533605341, 1.49265007423178537962110452318, 2.35810445145466476024325132745, 3.21657599996692471392800654820, 4.52890010262956596314025892402, 5.22971988558391834932662769370, 5.86890042022152039864705031678, 6.29234366362680868411565906656, 7.76636696498241087303353077965, 8.206662451910212620607517499213, 8.905574477027714282677781675335

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