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Properties

Label 2-354-177.176-c5-0-58
Degree $2$
Conductor $354$
Sign $0.0776 + 0.996i$
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 + (−14.5 − 5.59i)3-s + 16·4-s + 83.1i·5-s + (58.1 + 22.3i)6-s + 242.·7-s − 64·8-s + (180. + 162. i)9-s − 332. i·10-s − 603.·11-s + (−232. − 89.5i)12-s − 200. i·13-s − 970.·14-s + (465. − 1.21e3i)15-s + 256·16-s − 1.07e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.933 − 0.359i)3-s + 0.5·4-s + 1.48i·5-s + (0.659 + 0.254i)6-s + 1.87·7-s − 0.353·8-s + (0.741 + 0.670i)9-s − 1.05i·10-s − 1.50·11-s + (−0.466 − 0.179i)12-s − 0.329i·13-s − 1.32·14-s + (0.534 − 1.38i)15-s + 0.250·16-s − 0.905i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0776 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0776 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.0776 + 0.996i$
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: \((2,\ 354,\ (\ :5/2),\ 0.0776 + 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5912977854\)
\(L(\frac12)\) \(\approx\) \(0.5912977854\)
\(L(\frac{7}{2})\) 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 \( 1 + (14.5 + 5.59i)T \)
59 \( 1 + (-1.15e4 - 2.41e4i)T \)
good5 \( 1 - 83.1iT - 3.12e3T^{2} \)
7 \( 1 - 242.T + 1.68e4T^{2} \)
11 \( 1 + 603.T + 1.61e5T^{2} \)
13 \( 1 + 200. iT - 3.71e5T^{2} \)
17 \( 1 + 1.07e3iT - 1.41e6T^{2} \)
19 \( 1 + 306.T + 2.47e6T^{2} \)
23 \( 1 + 721.T + 6.43e6T^{2} \)
29 \( 1 - 1.17e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.63e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.80e3iT - 6.93e7T^{2} \)
41 \( 1 + 3.93e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.14e4T + 2.29e8T^{2} \)
53 \( 1 + 2.31e4iT - 4.18e8T^{2} \)
61 \( 1 + 3.68e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.05e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.91e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.44e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.64e4T + 3.07e9T^{2} \)
83 \( 1 + 6.21e4T + 3.93e9T^{2} \)
89 \( 1 - 9.49e4T + 5.58e9T^{2} \)
97 \( 1 - 1.74e4iT - 8.58e9T^{2} \)
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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

−10.61977725173616903273865906267, −9.903000909314862348507704547844, −8.009673393920394131962050857475, −7.76372765611707304177030990172, −6.79286918683906170552245799468, −5.64545605545716810257133995295, −4.72898524348142923918954226621, −2.73889242465016757814395341430, −1.77952790767150098587324943669, −0.24736772743891462261842927688, 1.02442984955952485903165724412, 1.88142391909489625191143679275, 4.27605751166445006801324977408, 5.07481649290998741010477269507, 5.67745759787748220864883702598, 7.34105049616007927689615799203, 8.309004003631041785830110528584, 8.769670962178375978574823613482, 10.14192935759656373851568660071, 10.77465454004733346888922737087

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