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Properties

Label 2-3420-95.64-c1-0-47
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  + (0.896 − 2.04i)5-s + 0.663i·7-s + 1.80·11-s + (−1.99 − 1.15i)13-s + (−3.77 + 2.18i)17-s + (−4.21 − 1.12i)19-s + (1.81 + 1.04i)23-s + (−3.39 − 3.67i)25-s + (−0.974 + 1.68i)29-s − 9.52·31-s + (1.35 + 0.594i)35-s + 2.97i·37-s + (0.247 + 0.428i)41-s + (−6.81 + 3.93i)43-s + (−5.69 − 3.28i)47-s + ⋯
L(s)  = 1  + (0.400 − 0.916i)5-s + 0.250i·7-s + 0.545·11-s + (−0.553 − 0.319i)13-s + (−0.915 + 0.528i)17-s + (−0.966 − 0.257i)19-s + (0.378 + 0.218i)23-s + (−0.678 − 0.734i)25-s + (−0.180 + 0.313i)29-s − 1.71·31-s + (0.229 + 0.100i)35-s + 0.489i·37-s + (0.0386 + 0.0669i)41-s + (−1.03 + 0.600i)43-s + (−0.830 − 0.479i)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: $\chi_{3420} (2629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.935 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06615795642\)
\(L(\frac12)\) \(\approx\) \(0.06615795642\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.896 + 2.04i)T \)
19 \( 1 + (4.21 + 1.12i)T \)
good7 \( 1 - 0.663iT - 7T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 + (1.99 + 1.15i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.77 - 2.18i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.81 - 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.974 - 1.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.52T + 31T^{2} \)
37 \( 1 - 2.97iT - 37T^{2} \)
41 \( 1 + (-0.247 - 0.428i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.81 - 3.93i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.69 + 3.28i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.99 - 1.15i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.88 + 6.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.36 - 9.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.96 + 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.95 - 5.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.86 + 2.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.99 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.20iT - 83T^{2} \)
89 \( 1 + (-6.65 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.80 - 5.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.408192431564348040667546563461, −7.46722258810176091953647295582, −6.60744382298752675382309111507, −5.91819501532843653413091478927, −5.07294167829593988312165187454, −4.45809432254633292685381010892, −3.51536811589362415350251206573, −2.27562222257575008897705696896, −1.50047469841648509281208629909, −0.01807700733738754745316277973, 1.76671907819412512418112789661, 2.49228968314370483766633091571, 3.55930711992724070747917384298, 4.30945930238195422663756132561, 5.27902272305871819342063548302, 6.17503271054881070201010752775, 6.87242153888181226475879608692, 7.26555248602912049058513799255, 8.274254957607577062822468478545, 9.209326164620696994871310872168

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