Properties

Label 2-354-177.176-c5-0-74
Degree $2$
Conductor $354$
Sign $0.724 + 0.689i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + (−3.72 + 15.1i)3-s + 16·4-s − 73.5i·5-s + (−14.8 + 60.5i)6-s + 107.·7-s + 64·8-s + (−215. − 112. i)9-s − 294. i·10-s − 36.3·11-s + (−59.5 + 242. i)12-s + 785. i·13-s + 431.·14-s + (1.11e3 + 273. i)15-s + 256·16-s − 1.63e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.238 + 0.971i)3-s + 0.5·4-s − 1.31i·5-s + (−0.168 + 0.686i)6-s + 0.831·7-s + 0.353·8-s + (−0.885 − 0.463i)9-s − 0.930i·10-s − 0.0905·11-s + (−0.119 + 0.485i)12-s + 1.28i·13-s + 0.588·14-s + (1.27 + 0.314i)15-s + 0.250·16-s − 1.37i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.724 + 0.689i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ 0.724 + 0.689i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.004783903\)
\(L(\frac12)\) \(\approx\) \(3.004783903\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 + (3.72 - 15.1i)T \)
59 \( 1 + (-1.32e4 + 2.32e4i)T \)
good5 \( 1 + 73.5iT - 3.12e3T^{2} \)
7 \( 1 - 107.T + 1.68e4T^{2} \)
11 \( 1 + 36.3T + 1.61e5T^{2} \)
13 \( 1 - 785. iT - 3.71e5T^{2} \)
17 \( 1 + 1.63e3iT - 1.41e6T^{2} \)
19 \( 1 + 509.T + 2.47e6T^{2} \)
23 \( 1 - 2.11e3T + 6.43e6T^{2} \)
29 \( 1 + 5.63e3iT - 2.05e7T^{2} \)
31 \( 1 - 359. iT - 2.86e7T^{2} \)
37 \( 1 + 1.00e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.65e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.92e3T + 2.29e8T^{2} \)
53 \( 1 - 4.06e3iT - 4.18e8T^{2} \)
61 \( 1 + 1.71e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.83e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.60e4iT - 1.80e9T^{2} \)
73 \( 1 + 6.09e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.99e4T + 3.07e9T^{2} \)
83 \( 1 - 4.20e4T + 3.93e9T^{2} \)
89 \( 1 - 1.42e5T + 5.58e9T^{2} \)
97 \( 1 - 1.42e5iT - 8.58e9T^{2} \)
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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

−10.75933337060094253406664073190, −9.428041288095974353558018949434, −8.933755694967099621774927492477, −7.77001920447339240888394918385, −6.37317469937663189378080969998, −4.99293934425770412616634858514, −4.87369856998738011373150464059, −3.81310247273994936208116799021, −2.16721763495892354799794924304, −0.64361896731065639042809814213, 1.27399336800218836983587223116, 2.49327252576438731715243408669, 3.41320480651889171249862546485, 5.04621433425155912149070855463, 6.00791300481229719217243646100, 6.83339021694456089115528781456, 7.69169444118557536434628142312, 8.469420982453460561079539850245, 10.47117520691803401884156050938, 10.81625936597635045999163270646

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