Properties

Label 2-3420-95.64-c1-0-45
Degree $2$
Conductor $3420$
Sign $-0.792 + 0.610i$
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  + (2.21 − 0.325i)5-s − 3.54i·7-s + 1.81·11-s + (−2.78 − 1.60i)13-s + (−6.92 + 3.99i)17-s + (0.863 + 4.27i)19-s + (−7.30 − 4.21i)23-s + (4.78 − 1.43i)25-s + (4.29 − 7.43i)29-s − 1.70·31-s + (−1.15 − 7.84i)35-s − 5.50i·37-s + (−4.05 − 7.02i)41-s + (−4.35 + 2.51i)43-s + (1.16 + 0.674i)47-s + ⋯
L(s)  = 1  + (0.989 − 0.145i)5-s − 1.34i·7-s + 0.547·11-s + (−0.771 − 0.445i)13-s + (−1.67 + 0.969i)17-s + (0.198 + 0.980i)19-s + (−1.52 − 0.878i)23-s + (0.957 − 0.287i)25-s + (0.796 − 1.38i)29-s − 0.306·31-s + (−0.194 − 1.32i)35-s − 0.905i·37-s + (−0.633 − 1.09i)41-s + (−0.663 + 0.383i)43-s + (0.170 + 0.0983i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.792 + 0.610i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.792 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.229071845\)
\(L(\frac12)\) \(\approx\) \(1.229071845\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.325i)T \)
19 \( 1 + (-0.863 - 4.27i)T \)
good7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + (2.78 + 1.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.92 - 3.99i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (7.30 + 4.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.29 + 7.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + 5.50iT - 37T^{2} \)
41 \( 1 + (4.05 + 7.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.35 - 2.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 0.674i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.92 + 1.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.960 + 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.04 + 4.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.94 - 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.82 + 1.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.08 - 3.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.30iT - 83T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.91 - 3.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.258443101013099866688874203085, −7.62456922042631335469926520195, −6.52396740442437791227669259814, −6.35145967539294700032855472333, −5.26953992224519706792419125214, −4.27925737308241366274684310250, −3.86142123103576156009278691559, −2.40854192454396165607609142295, −1.68118355274023560090132626409, −0.33013981208245476900717673737, 1.64343664695805878498200762594, 2.39445469119888632050164458574, 3.07684342715280196683442519208, 4.55797217011453390937193239333, 5.09064925910094507360130467090, 5.91734324792046595670164659830, 6.67552246924228014075554073105, 7.12829532064309513241381247403, 8.459516407999205574740463388166, 8.972503053914161277192973858843

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