Properties

Label 2-342-19.18-c2-0-6
Degree $2$
Conductor $342$
Sign $-i$
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  + 1.41i·2-s − 2.00·4-s + 5-s + 5·7-s − 2.82i·8-s + 1.41i·10-s − 5·11-s + 16.9i·13-s + 7.07i·14-s + 4.00·16-s + 25·17-s + 19·19-s − 2.00·20-s − 7.07i·22-s + 10·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + 0.200·5-s + 0.714·7-s − 0.353i·8-s + 0.141i·10-s − 0.454·11-s + 1.30i·13-s + 0.505i·14-s + 0.250·16-s + 1.47·17-s + 19-s − 0.100·20-s − 0.321i·22-s + 0.434·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-i$
Analytic conductor: \(9.31882\)
Root analytic conductor: \(3.05267\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20481 + 1.20481i\)
\(L(\frac12)\) \(\approx\) \(1.20481 + 1.20481i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
19 \( 1 - 19T \)
good5 \( 1 - T + 25T^{2} \)
7 \( 1 - 5T + 49T^{2} \)
11 \( 1 + 5T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 - 25T + 289T^{2} \)
23 \( 1 - 10T + 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 - 42.4iT - 961T^{2} \)
37 \( 1 - 25.4iT - 1.36e3T^{2} \)
41 \( 1 + 42.4iT - 1.68e3T^{2} \)
43 \( 1 - 5T + 1.84e3T^{2} \)
47 \( 1 + 5T + 2.20e3T^{2} \)
53 \( 1 + 25.4iT - 2.80e3T^{2} \)
59 \( 1 - 84.8iT - 3.48e3T^{2} \)
61 \( 1 - 95T + 3.72e3T^{2} \)
67 \( 1 + 110. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 + 42.4iT - 6.24e3T^{2} \)
83 \( 1 - 130T + 6.88e3T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 + 16.9iT - 9.40e3T^{2} \)
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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

−11.64064233313356387388896490604, −10.49635455337151594836612398517, −9.547610771394370137802218061984, −8.638209278007247935692835730783, −7.65496263507166528303072634816, −6.87194462039971213586716194650, −5.56134188790578949592862451624, −4.84652682556956400263962135709, −3.40476709370362647240071995959, −1.49924929012738049406820442972, 0.906681318693580108090346085312, 2.48976832081651316792378964185, 3.69898239721956865678086925630, 5.13025967508759334398046374998, 5.82265734008280160214125812000, 7.75262511500627942824781453916, 8.035420270137285824606144895634, 9.606460989148552492219919178808, 10.08031464678942435862383415381, 11.17899327868180286128228551186

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