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Properties

Label 2-3420-57.50-c1-0-0
Degree $2$
Conductor $3420$
Sign $-0.935 - 0.354i$
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 − 2.97·7-s − 2i·11-s + (3.37 + 1.94i)13-s + (2.48 − 1.43i)17-s + (−0.726 − 4.29i)19-s + (−1.90 − 1.10i)23-s + (0.499 − 0.866i)25-s + (−3.07 + 5.32i)29-s + 2.23i·31-s + (2.57 − 1.48i)35-s + 9.17i·37-s + (−0.713 − 1.23i)41-s + (−0.927 − 1.60i)43-s + (2.64 + 1.52i)47-s + ⋯
L(s)  = 1  + (−0.387 + 0.223i)5-s − 1.12·7-s − 0.603i·11-s + (0.936 + 0.540i)13-s + (0.602 − 0.347i)17-s + (−0.166 − 0.986i)19-s + (−0.398 − 0.229i)23-s + (0.0999 − 0.173i)25-s + (−0.570 + 0.988i)29-s + 0.402i·31-s + (0.435 − 0.251i)35-s + 1.50i·37-s + (−0.111 − 0.193i)41-s + (−0.141 − 0.245i)43-s + (0.385 + 0.222i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.935 - 0.354i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (2501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.935 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2500714461\)
\(L(\frac12)\) \(\approx\) \(0.2500714461\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.726 + 4.29i)T \)
good7 \( 1 + 2.97T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-3.37 - 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.48 + 1.43i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.90 + 1.10i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.07 - 5.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.23iT - 31T^{2} \)
37 \( 1 - 9.17iT - 37T^{2} \)
41 \( 1 + (0.713 + 1.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.927 + 1.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.64 - 1.52i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.86 + 4.96i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.35 - 4.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.26 - 5.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.99 + 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.22 + 7.31i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.38 - 9.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 - 5.99i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 + (-1.85 + 3.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.18 - 4.14i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.878168252417865238886641014663, −8.334070688588694796029918233076, −7.28469766746424078631309707998, −6.69883295084140911245709891927, −6.08538080336985001188038460663, −5.19471922279576657534974188552, −4.13563132099980502379623841748, −3.36941771332600296650140738185, −2.76314580563841814268545233566, −1.25928539439720377234405690661, 0.080736449241569530802610532932, 1.46021153380701507076391699801, 2.69550178425042088530740570722, 3.76600018715237653784695521904, 4.04268120822821445078879695772, 5.45653373878181359442484814428, 5.96476931881864406228839652910, 6.73014623854130741591371655301, 7.70731070454696510830677193983, 8.100670089094933174015600047050

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