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Properties

Label 2-354-3.2-c2-0-4
Degree $2$
Conductor $354$
Sign $0.165 - 0.986i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
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 + (−0.495 + 2.95i)3-s − 2.00·4-s − 8.96i·5-s + (4.18 + 0.701i)6-s − 9.25·7-s + 2.82i·8-s + (−8.50 − 2.93i)9-s − 12.6·10-s + 17.1i·11-s + (0.991 − 5.91i)12-s + 21.0·13-s + 13.0i·14-s + (26.5 + 4.44i)15-s + 4.00·16-s + 15.0i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.165 + 0.986i)3-s − 0.500·4-s − 1.79i·5-s + (0.697 + 0.116i)6-s − 1.32·7-s + 0.353i·8-s + (−0.945 − 0.325i)9-s − 1.26·10-s + 1.56i·11-s + (0.0826 − 0.493i)12-s + 1.61·13-s + 0.934i·14-s + (1.76 + 0.296i)15-s + 0.250·16-s + 0.886i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ 0.165 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.500836 + 0.423911i\)
\(L(\frac12)\) \(\approx\) \(0.500836 + 0.423911i\)
\(L(2)\) 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 \( 1 + (0.495 - 2.95i)T \)
59 \( 1 - 7.68iT \)
good5 \( 1 + 8.96iT - 25T^{2} \)
7 \( 1 + 9.25T + 49T^{2} \)
11 \( 1 - 17.1iT - 121T^{2} \)
13 \( 1 - 21.0T + 169T^{2} \)
17 \( 1 - 15.0iT - 289T^{2} \)
19 \( 1 + 11.0T + 361T^{2} \)
23 \( 1 - 38.9iT - 529T^{2} \)
29 \( 1 - 8.28iT - 841T^{2} \)
31 \( 1 - 31.3T + 961T^{2} \)
37 \( 1 + 30.2T + 1.36e3T^{2} \)
41 \( 1 - 27.3iT - 1.68e3T^{2} \)
43 \( 1 + 61.4T + 1.84e3T^{2} \)
47 \( 1 - 11.7iT - 2.20e3T^{2} \)
53 \( 1 + 14.5iT - 2.80e3T^{2} \)
61 \( 1 + 5.82T + 3.72e3T^{2} \)
67 \( 1 + 73.2T + 4.48e3T^{2} \)
71 \( 1 - 46.1iT - 5.04e3T^{2} \)
73 \( 1 - 85.4T + 5.32e3T^{2} \)
79 \( 1 + 13.7T + 6.24e3T^{2} \)
83 \( 1 - 32.5iT - 6.88e3T^{2} \)
89 \( 1 + 32.8iT - 7.92e3T^{2} \)
97 \( 1 + 88.8T + 9.40e3T^{2} \)
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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

−11.53820816579711583549299472316, −10.27895316863561942360157439284, −9.676865506695905961022677792628, −8.998277263623941778405043194927, −8.237732028065450244640766590881, −6.30734263987615198268739795304, −5.26790470735227465965449363678, −4.27378490772324755379342149569, −3.52675815968927942734303142446, −1.48015798966483168248125232937, 0.31623201792488229004870149551, 2.80094935154180547213937534173, 3.52384703881647592514543202191, 5.85861727358275072243093079550, 6.53527327273505567064148822049, 6.70712965771954565875536030037, 8.075653741154927801009604353937, 8.870405185221542331942860873906, 10.35255545271985579768533716325, 10.95535533058716492536817077540

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