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Properties

Label 2-354-177.176-c5-0-51
Degree $2$
Conductor $354$
Sign $-0.0265 - 0.999i$
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 + (−3.54 + 15.1i)3-s + 16·4-s + 29.8i·5-s + (−14.1 + 60.7i)6-s + 197.·7-s + 64·8-s + (−217. − 107. i)9-s + 119. i·10-s + 108.·11-s + (−56.6 + 242. i)12-s + 112. i·13-s + 791.·14-s + (−453. − 105. i)15-s + 256·16-s + 732. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.227 + 0.973i)3-s + 0.5·4-s + 0.534i·5-s + (−0.160 + 0.688i)6-s + 1.52·7-s + 0.353·8-s + (−0.896 − 0.442i)9-s + 0.377i·10-s + 0.270·11-s + (−0.113 + 0.486i)12-s + 0.184i·13-s + 1.07·14-s + (−0.520 − 0.121i)15-s + 0.250·16-s + 0.614i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0265 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0265 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.0265 - 0.999i$
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: \((2,\ 354,\ (\ :5/2),\ -0.0265 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.932779616\)
\(L(\frac12)\) \(\approx\) \(3.932779616\)
\(L(\frac{7}{2})\) 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 \( 1 + (3.54 - 15.1i)T \)
59 \( 1 + (2.58e4 - 6.76e3i)T \)
good5 \( 1 - 29.8iT - 3.12e3T^{2} \)
7 \( 1 - 197.T + 1.68e4T^{2} \)
11 \( 1 - 108.T + 1.61e5T^{2} \)
13 \( 1 - 112. iT - 3.71e5T^{2} \)
17 \( 1 - 732. iT - 1.41e6T^{2} \)
19 \( 1 - 3.04e3T + 2.47e6T^{2} \)
23 \( 1 - 135.T + 6.43e6T^{2} \)
29 \( 1 + 1.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.87e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.35e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.51e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.94e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.29e4T + 2.29e8T^{2} \)
53 \( 1 - 1.65e3iT - 4.18e8T^{2} \)
61 \( 1 - 5.26e3iT - 8.44e8T^{2} \)
67 \( 1 - 5.91e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.68e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.75e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.01e4T + 3.07e9T^{2} \)
83 \( 1 - 8.28e4T + 3.93e9T^{2} \)
89 \( 1 + 6.47e4T + 5.58e9T^{2} \)
97 \( 1 - 2.50e4iT - 8.58e9T^{2} \)
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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

−11.07538273482465185905212021441, −10.24766696729017186706100493680, −9.100388586818156699289667748206, −8.022527516627425510045730148863, −6.95446852559564619304025702150, −5.64336570174521691927785970488, −4.95839945134303323620016522797, −3.97709567537986480621679213497, −2.89839102928025606055534647347, −1.35768219409366915101212874369, 0.908188170453662652715299020779, 1.72205547678652671296520785432, 3.10250585527637615144147662164, 4.84933317048609688100826778126, 5.23366072719851072394614050959, 6.50343321471619831491719702744, 7.62557043903772732330287474576, 8.126413073554659807732409850827, 9.378589055219368101596477644613, 10.87789747199467283758647594979

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