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Properties

Label 2-3420-57.8-c1-0-16
Degree $2$
Conductor $3420$
Sign $0.967 - 0.251i$
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  + (0.866 + 0.5i)5-s + 3.67·7-s − 2i·11-s + (0.349 − 0.201i)13-s + (6.30 + 3.64i)17-s + (4.34 − 0.282i)19-s + (−6.22 + 3.59i)23-s + (0.499 + 0.866i)25-s + (3.94 + 6.84i)29-s − 8.62i·31-s + (3.18 + 1.83i)35-s + 1.01i·37-s + (4.14 − 7.17i)41-s + (−4.25 + 7.36i)43-s + (−6.28 + 3.63i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.223i)5-s + 1.38·7-s − 0.603i·11-s + (0.0969 − 0.0559i)13-s + (1.53 + 0.883i)17-s + (0.997 − 0.0649i)19-s + (−1.29 + 0.749i)23-s + (0.0999 + 0.173i)25-s + (0.733 + 1.27i)29-s − 1.54i·31-s + (0.537 + 0.310i)35-s + 0.166i·37-s + (0.647 − 1.12i)41-s + (−0.648 + 1.12i)43-s + (−0.917 + 0.529i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.967 - 0.251i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ 0.967 - 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.650011251\)
\(L(\frac12)\) \(\approx\) \(2.650011251\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-4.34 + 0.282i)T \)
good7 \( 1 - 3.67T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-0.349 + 0.201i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.30 - 3.64i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.22 - 3.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.94 - 6.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.62iT - 31T^{2} \)
37 \( 1 - 1.01iT - 37T^{2} \)
41 \( 1 + (-4.14 + 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.25 - 7.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.28 - 3.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.47 + 2.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.04 + 1.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.08 + 1.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.2 + 5.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.83 - 11.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.21 + 5.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.0 - 5.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.70iT - 83T^{2} \)
89 \( 1 + (-0.702 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.04 + 2.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.311313913016411081160261485377, −8.041541086424963299376910639590, −7.34518279227217201622375335896, −6.19918132872912455691133749661, −5.60039479105512329305581711837, −4.98255435397543227202562284533, −3.89373414406089698001184925896, −3.12662817583913982438559456012, −1.88426135566742820579000636669, −1.13361723541321423828866317958, 0.995523409781647763520181316353, 1.86503073287295623449434724834, 2.87756572872208216863075156047, 4.02887535014128069877180487197, 4.95388859014949327328178036597, 5.29523806398668772653210656194, 6.28634602302931689886504850585, 7.22545862998780897769917067947, 7.971287176855845407758332751036, 8.329930039739693722246787634030

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