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Properties

Label 2-354-177.176-c5-0-21
Degree $2$
Conductor $354$
Sign $0.913 - 0.407i$
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 + (3.22 − 15.2i)3-s + 16·4-s − 58.3i·5-s + (−12.8 + 61.0i)6-s + 20.4·7-s − 64·8-s + (−222. − 98.3i)9-s + 233. i·10-s + 586.·11-s + (51.5 − 244. i)12-s + 297. i·13-s − 81.8·14-s + (−890. − 188. i)15-s + 256·16-s + 1.71e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.206 − 0.978i)3-s + 0.5·4-s − 1.04i·5-s + (−0.146 + 0.691i)6-s + 0.157·7-s − 0.353·8-s + (−0.914 − 0.404i)9-s + 0.738i·10-s + 1.46·11-s + (0.103 − 0.489i)12-s + 0.488i·13-s − 0.111·14-s + (−1.02 − 0.215i)15-s + 0.250·16-s + 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.913 - 0.407i$
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: \((2,\ 354,\ (\ :5/2),\ 0.913 - 0.407i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.173780662\)
\(L(\frac12)\) \(\approx\) \(1.173780662\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.22 + 15.2i)T \)
59 \( 1 + (1.56e4 + 2.16e4i)T \)
good5 \( 1 + 58.3iT - 3.12e3T^{2} \)
7 \( 1 - 20.4T + 1.68e4T^{2} \)
11 \( 1 - 586.T + 1.61e5T^{2} \)
13 \( 1 - 297. iT - 3.71e5T^{2} \)
17 \( 1 - 1.71e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.16e3T + 2.47e6T^{2} \)
23 \( 1 - 1.60e3T + 6.43e6T^{2} \)
29 \( 1 - 7.95e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.71e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.05e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.06e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.49e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.91e4T + 2.29e8T^{2} \)
53 \( 1 - 4.60e3iT - 4.18e8T^{2} \)
61 \( 1 + 345. iT - 8.44e8T^{2} \)
67 \( 1 + 5.45e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.20e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.14e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.70e4T + 3.07e9T^{2} \)
83 \( 1 - 8.00e4T + 3.93e9T^{2} \)
89 \( 1 - 3.43e4T + 5.58e9T^{2} \)
97 \( 1 + 5.13e4iT - 8.58e9T^{2} \)
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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

−10.79453464628499722578369221317, −9.371789859268896164722346354453, −8.702836989409378675732008768228, −8.221489780696454239661167618935, −6.84908994692205756013481905909, −6.31459094649530235249052069447, −4.83230530716451166630090809021, −3.38005866325802031761033518712, −1.55001134955952400691482420960, −1.29605249411563796387134039026, 0.38543329488778381557810393004, 2.28180691080551254371412681057, 3.29676538097568306294652209857, 4.41620266602793723648646017287, 5.88798896815829102531316006716, 6.84962990179424142338506428904, 7.86921274204469224251752867612, 8.999650498308818068437328542248, 9.627990861581646770113347576346, 10.46267770290677402597371553581

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