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Properties

Label 2-354-177.176-c5-0-68
Degree $2$
Conductor $354$
Sign $0.995 - 0.0990i$
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 + (15.2 + 3.38i)3-s + 16·4-s − 48.6i·5-s + (60.8 + 13.5i)6-s + 135.·7-s + 64·8-s + (220. + 103. i)9-s − 194. i·10-s − 414.·11-s + (243. + 54.1i)12-s + 781. i·13-s + 541.·14-s + (164. − 740. i)15-s + 256·16-s + 1.01e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.976 + 0.217i)3-s + 0.5·4-s − 0.870i·5-s + (0.690 + 0.153i)6-s + 1.04·7-s + 0.353·8-s + (0.905 + 0.424i)9-s − 0.615i·10-s − 1.03·11-s + (0.488 + 0.108i)12-s + 1.28i·13-s + 0.739·14-s + (0.189 − 0.849i)15-s + 0.250·16-s + 0.850i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.995 - 0.0990i$
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: \((2,\ 354,\ (\ :5/2),\ 0.995 - 0.0990i)\)

Particular Values

\(L(3)\) \(\approx\) \(5.664032949\)
\(L(\frac12)\) \(\approx\) \(5.664032949\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.2 - 3.38i)T \)
59 \( 1 + (-2.53e4 + 8.36e3i)T \)
good5 \( 1 + 48.6iT - 3.12e3T^{2} \)
7 \( 1 - 135.T + 1.68e4T^{2} \)
11 \( 1 + 414.T + 1.61e5T^{2} \)
13 \( 1 - 781. iT - 3.71e5T^{2} \)
17 \( 1 - 1.01e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.89e3T + 2.47e6T^{2} \)
23 \( 1 - 2.11e3T + 6.43e6T^{2} \)
29 \( 1 + 595. iT - 2.05e7T^{2} \)
31 \( 1 - 222. iT - 2.86e7T^{2} \)
37 \( 1 + 7.79e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.22e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.28e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.10e3T + 2.29e8T^{2} \)
53 \( 1 + 3.98e4iT - 4.18e8T^{2} \)
61 \( 1 + 1.11e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.87e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.54e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.17e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.24e4T + 3.07e9T^{2} \)
83 \( 1 + 3.31e4T + 3.93e9T^{2} \)
89 \( 1 + 4.74e4T + 5.58e9T^{2} \)
97 \( 1 - 3.70e4iT - 8.58e9T^{2} \)
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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

−10.76719470173296887141718460877, −9.620465374945692586121456216872, −8.658898925155866998922866206540, −7.950323101846276890854129050770, −6.99363835412127615671697954479, −5.29800239937875686018270043751, −4.72197777081289625170043397403, −3.68504320342877247937018192916, −2.29103621309158936986230812280, −1.33050472256956448656933711787, 1.17869804878197459815648693543, 2.74928354875390997272107708330, 3.06408079190142105470503400095, 4.66469222041596125066426331801, 5.56861033267288769937256479609, 7.11657899052525273045056372325, 7.60447128574625001399577061043, 8.491863308932599928819269896378, 9.888376642814818590295494055625, 10.68971360842355647058865409939

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