Properties

Label 2-342-19.8-c2-0-9
Degree $2$
Conductor $342$
Sign $0.516 - 0.856i$
Analytic cond. $9.31882$
Root an. cond. $3.05267$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−0.5 + 0.866i)5-s + 6.89·7-s + 2.82i·8-s + (−1.22 + 0.707i)10-s + 14.8·11-s + (−14.8 + 8.57i)13-s + (8.44 + 4.87i)14-s + (−2.00 + 3.46i)16-s + (−1.05 + 1.81i)17-s + (11.3 − 15.2i)19-s − 1.99·20-s + (18.2 + 10.5i)22-s + (13.5 + 23.4i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.100 + 0.173i)5-s + 0.985·7-s + 0.353i·8-s + (−0.122 + 0.0707i)10-s + 1.35·11-s + (−1.14 + 0.659i)13-s + (0.603 + 0.348i)14-s + (−0.125 + 0.216i)16-s + (−0.0617 + 0.107i)17-s + (0.597 − 0.802i)19-s − 0.0999·20-s + (0.829 + 0.478i)22-s + (0.587 + 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(9.31882\)
Root analytic conductor: \(3.05267\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1),\ 0.516 - 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27162 + 1.28347i\)
\(L(\frac12)\) \(\approx\) \(2.27162 + 1.28347i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
19 \( 1 + (-11.3 + 15.2i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 6.89T + 49T^{2} \)
11 \( 1 - 14.8T + 121T^{2} \)
13 \( 1 + (14.8 - 8.57i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (1.05 - 1.81i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-13.5 - 23.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.54 + 3.20i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 31.1iT - 961T^{2} \)
37 \( 1 - 28.9iT - 1.36e3T^{2} \)
41 \( 1 + (55.9 + 32.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-37.6 + 65.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (5.77 + 9.99i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-69.2 + 40.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (50.9 + 29.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.09 + 1.89i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.6 - 29.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (87.5 + 50.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-63.6 + 110. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (5.78 + 3.33i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 1.30T + 6.88e3T^{2} \)
89 \( 1 + (5.84 - 3.37i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (114. + 65.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.77639010029787156626668247923, −10.80339924207866382551551320104, −9.429605564372680415952366309431, −8.641731697901507680483566419560, −7.28575784303945273367405535974, −6.86435633091017572108956728054, −5.34396036968554456814811953619, −4.58319153036334181551886575365, −3.33465407464470554508804491632, −1.68805169903914302542831955799, 1.17711945985184569637571064324, 2.67912080795435577581166830446, 4.17464443521666457191145373090, 4.95030939050624974758213535146, 6.12309122860035375677078677456, 7.29628789764897497157384690698, 8.310185443960419854759449740819, 9.434208504948894955522535249406, 10.38143856696529440768279186459, 11.38163693824129740217512025106

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