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Properties

Label 2-342-19.6-c1-0-4
Degree $2$
Conductor $342$
Sign $0.854 + 0.519i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.386 + 2.19i)5-s + (1.32 − 2.29i)7-s + (−0.500 − 0.866i)8-s + (1.70 + 1.43i)10-s + (1.11 + 1.92i)11-s + (4.97 − 1.80i)13-s + (−0.460 − 2.61i)14-s + (−0.939 − 0.342i)16-s + (2.61 − 2.19i)17-s + (−4.29 + 0.725i)19-s + 2.22·20-s + (2.09 + 0.761i)22-s + (−0.386 + 2.19i)23-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (0.172 + 0.980i)5-s + (0.501 − 0.868i)7-s + (−0.176 − 0.306i)8-s + (0.539 + 0.452i)10-s + (0.335 + 0.581i)11-s + (1.37 − 0.501i)13-s + (−0.123 − 0.698i)14-s + (−0.234 − 0.0855i)16-s + (0.633 − 0.531i)17-s + (−0.986 + 0.166i)19-s + 0.497·20-s + (0.446 + 0.162i)22-s + (−0.0806 + 0.457i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.854 + 0.519i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86794 - 0.523122i\)
\(L(\frac12)\) \(\approx\) \(1.86794 - 0.523122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 \)
19 \( 1 + (4.29 - 0.725i)T \)
good5 \( 1 + (-0.386 - 2.19i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.32 + 2.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.97 + 1.80i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.61 + 2.19i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.386 - 2.19i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.68 + 3.09i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.15 - 8.93i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + (6.79 + 2.47i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.02 + 5.83i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.43 + 7.07i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.70 - 9.67i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (3.79 - 3.18i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.990 - 5.61i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.56 - 5.51i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.764 + 4.33i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.62 - 0.956i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.9 + 4.72i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.25 - 9.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.34 + 2.67i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (13.6 - 11.4i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.26531591348235476960981126017, −10.65837699752636919040530250501, −10.04001102548875137609867240011, −8.684170507505647427662981945641, −7.41493359783987197129705917966, −6.60956858322865786724205007400, −5.47713537837868240985206338339, −4.12487797305253992391178681444, −3.22278419973998519856952541815, −1.59764659408045351461024667352, 1.73055386427953094767451718239, 3.59347273304113576388481095578, 4.73171287352888648887954841038, 5.74977619614352700463125573620, 6.43616766283468013804228892744, 8.114140975945802718124571396868, 8.589307515440990334973628089047, 9.420607718868038044859973413939, 11.05346428581987545004312603305, 11.62888413245839038589165976910

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