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Properties

Label 2-354-177.176-c3-0-22
Degree $2$
Conductor $354$
Sign $0.0116 - 0.999i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (1.53 + 4.96i)3-s + 4·4-s + 1.00i·5-s + (−3.06 − 9.93i)6-s + 8.17·7-s − 8·8-s + (−22.3 + 15.1i)9-s − 2.01i·10-s + 49.9·11-s + (6.12 + 19.8i)12-s − 46.3i·13-s − 16.3·14-s + (−5.01 + 1.54i)15-s + 16·16-s + 111. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.294 + 0.955i)3-s + 0.5·4-s + 0.0903i·5-s + (−0.208 − 0.675i)6-s + 0.441·7-s − 0.353·8-s + (−0.826 + 0.562i)9-s − 0.0638i·10-s + 1.36·11-s + (0.147 + 0.477i)12-s − 0.988i·13-s − 0.311·14-s + (−0.0863 + 0.0265i)15-s + 0.250·16-s + 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0116 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0116 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.0116 - 0.999i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.0116 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.619111975\)
\(L(\frac12)\) \(\approx\) \(1.619111975\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (-1.53 - 4.96i)T \)
59 \( 1 + (431. + 138. i)T \)
good5 \( 1 - 1.00iT - 125T^{2} \)
7 \( 1 - 8.17T + 343T^{2} \)
11 \( 1 - 49.9T + 1.33e3T^{2} \)
13 \( 1 + 46.3iT - 2.19e3T^{2} \)
17 \( 1 - 111. iT - 4.91e3T^{2} \)
19 \( 1 - 48.6T + 6.85e3T^{2} \)
23 \( 1 - 73.0T + 1.21e4T^{2} \)
29 \( 1 - 10.9iT - 2.43e4T^{2} \)
31 \( 1 - 14.5iT - 2.97e4T^{2} \)
37 \( 1 + 2.36iT - 5.06e4T^{2} \)
41 \( 1 - 127. iT - 6.89e4T^{2} \)
43 \( 1 + 82.9iT - 7.95e4T^{2} \)
47 \( 1 + 39.4T + 1.03e5T^{2} \)
53 \( 1 - 434. iT - 1.48e5T^{2} \)
61 \( 1 - 572. iT - 2.26e5T^{2} \)
67 \( 1 - 265. iT - 3.00e5T^{2} \)
71 \( 1 - 491. iT - 3.57e5T^{2} \)
73 \( 1 - 841. iT - 3.89e5T^{2} \)
79 \( 1 - 1.30e3T + 4.93e5T^{2} \)
83 \( 1 + 322.T + 5.71e5T^{2} \)
89 \( 1 - 816.T + 7.04e5T^{2} \)
97 \( 1 + 854. iT - 9.12e5T^{2} \)
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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

−10.93633035716520524647652464913, −10.36614809696284965257197121834, −9.356059919313322008963220601984, −8.639133381320175170685576873163, −7.83777713355875166943679623074, −6.51505379226207147760136362450, −5.38661730935726682795238347308, −4.08066762610599248133721583494, −2.99370465836249049528840518511, −1.30681561991428764801515707671, 0.800565401707553899244809592560, 1.85660546783598468972770409481, 3.24722036554883644424430827250, 4.92120569654893354207299458685, 6.47198872555339419130597526532, 7.02520894879351070734773863761, 7.975844132853637485215941775665, 9.166430246034370617522555711019, 9.324062789062615498470897071249, 11.06758147656928440936211163607

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