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Properties

Label 2-354-177.176-c5-0-20
Degree $2$
Conductor $354$
Sign $-0.997 - 0.0698i$
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 + (−9.68 + 12.2i)3-s + 16·4-s + 48.6i·5-s + (38.7 − 48.8i)6-s + 149.·7-s − 64·8-s + (−55.5 − 236. i)9-s − 194. i·10-s + 118.·11-s + (−154. + 195. i)12-s + 964. i·13-s − 596.·14-s + (−593. − 470. i)15-s + 256·16-s − 1.57e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.621 + 0.783i)3-s + 0.5·4-s + 0.869i·5-s + (0.439 − 0.554i)6-s + 1.15·7-s − 0.353·8-s + (−0.228 − 0.973i)9-s − 0.614i·10-s + 0.294·11-s + (−0.310 + 0.391i)12-s + 1.58i·13-s − 0.813·14-s + (−0.681 − 0.540i)15-s + 0.250·16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.997 - 0.0698i$
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: \((2,\ 354,\ (\ :5/2),\ -0.997 - 0.0698i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9290331839\)
\(L(\frac12)\) \(\approx\) \(0.9290331839\)
\(L(\frac{7}{2})\) 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 \( 1 + (9.68 - 12.2i)T \)
59 \( 1 + (1.51e4 + 2.20e4i)T \)
good5 \( 1 - 48.6iT - 3.12e3T^{2} \)
7 \( 1 - 149.T + 1.68e4T^{2} \)
11 \( 1 - 118.T + 1.61e5T^{2} \)
13 \( 1 - 964. iT - 3.71e5T^{2} \)
17 \( 1 + 1.57e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.97e3T + 2.47e6T^{2} \)
23 \( 1 - 3.40e3T + 6.43e6T^{2} \)
29 \( 1 - 4.80e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.75e3iT - 2.86e7T^{2} \)
37 \( 1 - 5.99e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.42e3iT - 1.15e8T^{2} \)
43 \( 1 - 6.93e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.61e4T + 2.29e8T^{2} \)
53 \( 1 - 7.64e3iT - 4.18e8T^{2} \)
61 \( 1 + 1.22e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.83e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.43e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.02e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.84e4T + 3.07e9T^{2} \)
83 \( 1 + 1.62e4T + 3.93e9T^{2} \)
89 \( 1 - 2.63e4T + 5.58e9T^{2} \)
97 \( 1 + 4.29e4iT - 8.58e9T^{2} \)
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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

−11.05527453492919563611933133657, −10.38803489218280457069522625663, −9.159394074260556134312781787691, −8.667986577455361339164893704397, −7.01530131861316169434160838473, −6.66984308174450218400605165240, −5.13632927370569771488492211398, −4.23975062299538763615038125623, −2.74037695214449764111593501263, −1.32056739257965632959511185040, 0.36724417849392656639187296145, 1.25070236707899449713832454403, 2.28702239770647020038333096733, 4.36683892630389908152183618032, 5.47536102271373687011771695285, 6.31213615941361197694190121560, 7.68702319845621689402305889640, 8.201806483963808499008984373202, 8.946289408416913550597966950290, 10.61190383799473541870975487970

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