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Properties

Label 2-354-177.176-c5-0-40
Degree $2$
Conductor $354$
Sign $0.634 - 0.772i$
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 + (13.8 − 7.23i)3-s + 16·4-s + 44.6i·5-s + (−55.2 + 28.9i)6-s + 154.·7-s − 64·8-s + (138. − 199. i)9-s − 178. i·10-s − 566.·11-s + (220. − 115. i)12-s + 59.6i·13-s − 617.·14-s + (322. + 616. i)15-s + 256·16-s + 1.84e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.885 − 0.464i)3-s + 0.5·4-s + 0.798i·5-s + (−0.626 + 0.328i)6-s + 1.19·7-s − 0.353·8-s + (0.569 − 0.822i)9-s − 0.564i·10-s − 1.41·11-s + (0.442 − 0.232i)12-s + 0.0978i·13-s − 0.841·14-s + (0.370 + 0.707i)15-s + 0.250·16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.634 - 0.772i$
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: \((2,\ 354,\ (\ :5/2),\ 0.634 - 0.772i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.158311045\)
\(L(\frac12)\) \(\approx\) \(2.158311045\)
\(L(\frac{7}{2})\) 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 \( 1 + (-13.8 + 7.23i)T \)
59 \( 1 + (2.46e4 - 1.04e4i)T \)
good5 \( 1 - 44.6iT - 3.12e3T^{2} \)
7 \( 1 - 154.T + 1.68e4T^{2} \)
11 \( 1 + 566.T + 1.61e5T^{2} \)
13 \( 1 - 59.6iT - 3.71e5T^{2} \)
17 \( 1 - 1.84e3iT - 1.41e6T^{2} \)
19 \( 1 + 564.T + 2.47e6T^{2} \)
23 \( 1 - 4.50e3T + 6.43e6T^{2} \)
29 \( 1 + 2.13e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.56e3iT - 2.86e7T^{2} \)
37 \( 1 - 3.24e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.95e4iT - 1.15e8T^{2} \)
43 \( 1 - 5.28e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.39e3T + 2.29e8T^{2} \)
53 \( 1 + 2.02e4iT - 4.18e8T^{2} \)
61 \( 1 + 2.33e4iT - 8.44e8T^{2} \)
67 \( 1 - 5.25e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.66e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.42e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.50e4T + 3.07e9T^{2} \)
83 \( 1 + 1.27e4T + 3.93e9T^{2} \)
89 \( 1 - 2.97e4T + 5.58e9T^{2} \)
97 \( 1 - 8.09e4iT - 8.58e9T^{2} \)
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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

−10.73448453438102687174026119943, −9.876287283959338651243629957849, −8.555169208787494394358771786787, −8.116170937996401960479414717300, −7.28901779015084963702360952664, −6.32199429481670710049846539395, −4.80832415980482507235642076851, −3.21088069913079228988590576223, −2.28232980383529491231813252867, −1.21898493026534064908319828218, 0.65352554806960367943119887400, 1.99591822584108610144668116625, 3.03545330916275418399259019037, 4.79875428104034585493195196641, 5.17188632743367108705671360860, 7.28335025163393095462490135720, 7.84908057515993324891808026707, 8.813221745641077719267631427954, 9.232089413655255387958790471826, 10.55094992060192222809439826865

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