Properties

Label 2-3420-95.49-c1-0-13
Degree $2$
Conductor $3420$
Sign $-0.0610 - 0.998i$
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  + (−2.22 + 0.248i)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 + (4.87 − 1.10i)25-s + (−0.974 − 1.68i)29-s − 9.52·31-s + (−0.164 − 1.47i)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  + (−0.993 + 0.111i)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 + (0.975 − 0.220i)25-s + (−0.180 − 0.313i)29-s − 1.71·31-s + (−0.0278 − 0.249i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3420\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.0610 - 0.998i$
Analytic conductor: \(27.3088\)
Root analytic conductor: \(5.22578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3420} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3420,\ (\ :1/2),\ -0.0610 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.116328025\)
\(L(\frac12)\) \(\approx\) \(1.116328025\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.248i)T \)
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

−8.820123072000449543317053677227, −7.86829403749601438128329623727, −7.59361620816706626694087337576, −6.47416433130303587888575777996, −5.90879447886989893316521483101, −4.94623529292305885878783201336, −3.84054299233061269868659989498, −3.61960681378035281756281256868, −2.31515724558105163366369896108, −1.08334459409127265988485145723, 0.39530416323670179525232475505, 1.61967349950890523360830099226, 2.92985609932368002426655743281, 3.91096023644377098435339831239, 4.26344889890778354789363771158, 5.38994237050112824144213814648, 6.16751044688723853281090393653, 7.23706969772370025922816815037, 7.42949786338315430882292355128, 8.550825132510842818772700136237

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