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Properties

Label 2-354-177.176-c5-0-23
Degree $2$
Conductor $354$
Sign $-0.169 - 0.985i$
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 + (−13.3 − 8.07i)3-s + 16·4-s + 83.4i·5-s + (53.3 + 32.3i)6-s − 123.·7-s − 64·8-s + (112. + 215. i)9-s − 333. i·10-s − 49.5·11-s + (−213. − 129. i)12-s − 349. i·13-s + 495.·14-s + (674. − 1.11e3i)15-s + 256·16-s + 992. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.855 − 0.518i)3-s + 0.5·4-s + 1.49i·5-s + (0.604 + 0.366i)6-s − 0.955·7-s − 0.353·8-s + (0.463 + 0.886i)9-s − 1.05i·10-s − 0.123·11-s + (−0.427 − 0.259i)12-s − 0.573i·13-s + 0.675·14-s + (0.773 − 1.27i)15-s + 0.250·16-s + 0.833i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.169 - 0.985i$
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: \((2,\ 354,\ (\ :5/2),\ -0.169 - 0.985i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7551182186\)
\(L(\frac12)\) \(\approx\) \(0.7551182186\)
\(L(\frac{7}{2})\) 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 \( 1 + (13.3 + 8.07i)T \)
59 \( 1 + (1.75e4 + 2.01e4i)T \)
good5 \( 1 - 83.4iT - 3.12e3T^{2} \)
7 \( 1 + 123.T + 1.68e4T^{2} \)
11 \( 1 + 49.5T + 1.61e5T^{2} \)
13 \( 1 + 349. iT - 3.71e5T^{2} \)
17 \( 1 - 992. iT - 1.41e6T^{2} \)
19 \( 1 - 2.08e3T + 2.47e6T^{2} \)
23 \( 1 - 3.91e3T + 6.43e6T^{2} \)
29 \( 1 + 1.00e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.15e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.57e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.21e4iT - 1.15e8T^{2} \)
43 \( 1 - 437. iT - 1.47e8T^{2} \)
47 \( 1 - 2.40e4T + 2.29e8T^{2} \)
53 \( 1 - 1.80e4iT - 4.18e8T^{2} \)
61 \( 1 + 2.68e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.93e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.51e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.72e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.15e4T + 3.07e9T^{2} \)
83 \( 1 - 9.51e4T + 3.93e9T^{2} \)
89 \( 1 - 1.21e5T + 5.58e9T^{2} \)
97 \( 1 - 4.92e4iT - 8.58e9T^{2} \)
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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

−10.66936510565057242764984065766, −10.33325665966237743537215352910, −9.225434619015901320171917411079, −7.72190649142781409673548539883, −7.08417444694742446816959392584, −6.36660279086355802713706583779, −5.47201710632319913577388588960, −3.45343451861089692194178983586, −2.49248626992632462727012947601, −0.897180591603532655923661235377, 0.40678874577510432866161011292, 1.19401348795618387150041588031, 3.19797700472461347446049192065, 4.64179879300377212163250441864, 5.41104360210269114913008659919, 6.53408800379237224561382171792, 7.54516857924052415317349485504, 9.139416004482535574692012846964, 9.207087078160536691863991233394, 10.17409047667852617386495895770

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