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Properties

Label 2-354-177.176-c5-0-41
Degree $2$
Conductor $354$
Sign $0.993 + 0.116i$
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 + (−9.75 + 12.1i)3-s + 16·4-s − 43.2i·5-s + (39.0 − 48.6i)6-s − 5.13·7-s − 64·8-s + (−52.5 − 237. i)9-s + 173. i·10-s + 262.·11-s + (−156. + 194. i)12-s + 296. i·13-s + 20.5·14-s + (526. + 422. i)15-s + 256·16-s + 1.33e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.625 + 0.779i)3-s + 0.5·4-s − 0.774i·5-s + (0.442 − 0.551i)6-s − 0.0396·7-s − 0.353·8-s + (−0.216 − 0.976i)9-s + 0.547i·10-s + 0.654·11-s + (−0.312 + 0.389i)12-s + 0.485i·13-s + 0.0280·14-s + (0.603 + 0.484i)15-s + 0.250·16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.993 + 0.116i$
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: \((2,\ 354,\ (\ :5/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9230549752\)
\(L(\frac12)\) \(\approx\) \(0.9230549752\)
\(L(\frac{7}{2})\) 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 \( 1 + (9.75 - 12.1i)T \)
59 \( 1 + (-1.41e4 - 2.26e4i)T \)
good5 \( 1 + 43.2iT - 3.12e3T^{2} \)
7 \( 1 + 5.13T + 1.68e4T^{2} \)
11 \( 1 - 262.T + 1.61e5T^{2} \)
13 \( 1 - 296. iT - 3.71e5T^{2} \)
17 \( 1 - 1.33e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.34e3T + 2.47e6T^{2} \)
23 \( 1 + 1.85e3T + 6.43e6T^{2} \)
29 \( 1 - 1.12e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.10e3iT - 2.86e7T^{2} \)
37 \( 1 + 6.96e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.64e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.56e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.35e4T + 2.29e8T^{2} \)
53 \( 1 - 1.28e4iT - 4.18e8T^{2} \)
61 \( 1 + 5.07e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.30e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.76e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.63e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.66e4T + 3.07e9T^{2} \)
83 \( 1 - 5.27e4T + 3.93e9T^{2} \)
89 \( 1 - 8.97e4T + 5.58e9T^{2} \)
97 \( 1 + 2.49e4iT - 8.58e9T^{2} \)
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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

−10.56852608204178370525887251364, −9.624485640102907333105671618033, −8.954336720631102687656474469349, −8.117712623882011177131668438038, −6.63156721740913847694382102939, −5.89266401736150462722082264808, −4.61023167669503409017357519738, −3.71826321027975603976349672484, −1.82058874970956294304094733174, −0.52029369055624593796104724267, 0.68726993755061764969144205939, 1.97959053193522699995543372878, 3.15413819488660817937714272664, 4.94443252619524521290715771032, 6.28467515719432217770175413290, 6.79929468400384591951784542280, 7.73239813660815889192974595866, 8.671454468858712424298179281165, 9.916259632755676708499011688204, 10.69252714369056615596167708520

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