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Properties

Label 2-354-177.176-c3-0-29
Degree $2$
Conductor $354$
Sign $-0.403 + 0.915i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (−4.86 − 1.82i)3-s + 4·4-s − 6.14i·5-s + (9.72 + 3.65i)6-s − 15.5·7-s − 8·8-s + (20.3 + 17.7i)9-s + 12.2i·10-s + 37.5·11-s + (−19.4 − 7.31i)12-s − 29.6i·13-s + 31.0·14-s + (−11.2 + 29.8i)15-s + 16·16-s + 47.7i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.936 − 0.351i)3-s + 0.5·4-s − 0.549i·5-s + (0.661 + 0.248i)6-s − 0.837·7-s − 0.353·8-s + (0.752 + 0.658i)9-s + 0.388i·10-s + 1.03·11-s + (−0.468 − 0.175i)12-s − 0.632i·13-s + 0.592·14-s + (−0.193 + 0.514i)15-s + 0.250·16-s + 0.681i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.403 + 0.915i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ -0.403 + 0.915i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6747561671\)
\(L(\frac12)\) \(\approx\) \(0.6747561671\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (4.86 + 1.82i)T \)
59 \( 1 + (-25.0 + 452. i)T \)
good5 \( 1 + 6.14iT - 125T^{2} \)
7 \( 1 + 15.5T + 343T^{2} \)
11 \( 1 - 37.5T + 1.33e3T^{2} \)
13 \( 1 + 29.6iT - 2.19e3T^{2} \)
17 \( 1 - 47.7iT - 4.91e3T^{2} \)
19 \( 1 - 69.7T + 6.85e3T^{2} \)
23 \( 1 + 3.79T + 1.21e4T^{2} \)
29 \( 1 - 129. iT - 2.43e4T^{2} \)
31 \( 1 + 12.3iT - 2.97e4T^{2} \)
37 \( 1 + 237. iT - 5.06e4T^{2} \)
41 \( 1 - 336. iT - 6.89e4T^{2} \)
43 \( 1 + 347. iT - 7.95e4T^{2} \)
47 \( 1 + 173.T + 1.03e5T^{2} \)
53 \( 1 + 99.3iT - 1.48e5T^{2} \)
61 \( 1 + 764. iT - 2.26e5T^{2} \)
67 \( 1 + 596. iT - 3.00e5T^{2} \)
71 \( 1 + 755. iT - 3.57e5T^{2} \)
73 \( 1 + 360. iT - 3.89e5T^{2} \)
79 \( 1 + 963.T + 4.93e5T^{2} \)
83 \( 1 + 669.T + 5.71e5T^{2} \)
89 \( 1 - 337.T + 7.04e5T^{2} \)
97 \( 1 - 765. iT - 9.12e5T^{2} \)
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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

−10.72850811485309832935002665542, −9.828618079646785852482759540068, −9.018031944237262965128929030287, −7.888868030917485508131300011739, −6.83587152175607332135226061386, −6.11189710843640241654990798471, −5.00566405347742426890997646590, −3.45643733571945333956145736709, −1.55481975663707001915529286580, −0.42044281230418058877634863789, 1.10376664937780530868071775478, 3.01355503971671039796516152714, 4.28665576689219484100080529446, 5.77184118193747769070410036307, 6.69105990372411132552573191484, 7.20083491169422993360815866294, 8.860376694189981279810848909767, 9.686446925963523068059114861728, 10.22369301164868092233976493226, 11.47494297971548896581403191062

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