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Properties

Label 2-354-177.176-c5-0-96
Degree $2$
Conductor $354$
Sign $-0.834 + 0.551i$
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 + (12.4 − 9.34i)3-s + 16·4-s − 71.5i·5-s + (49.8 − 37.3i)6-s − 19.6·7-s + 64·8-s + (68.1 − 233. i)9-s − 286. i·10-s − 260.·11-s + (199. − 149. i)12-s − 65.1i·13-s − 78.4·14-s + (−669. − 892. i)15-s + 256·16-s − 446. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.800 − 0.599i)3-s + 0.5·4-s − 1.28i·5-s + (0.565 − 0.424i)6-s − 0.151·7-s + 0.353·8-s + (0.280 − 0.959i)9-s − 0.905i·10-s − 0.649·11-s + (0.400 − 0.299i)12-s − 0.106i·13-s − 0.107·14-s + (−0.767 − 1.02i)15-s + 0.250·16-s − 0.374i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.834 + 0.551i$
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: \((2,\ 354,\ (\ :5/2),\ -0.834 + 0.551i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.606096890\)
\(L(\frac12)\) \(\approx\) \(3.606096890\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12.4 + 9.34i)T \)
59 \( 1 + (2.66e4 + 1.57e3i)T \)
good5 \( 1 + 71.5iT - 3.12e3T^{2} \)
7 \( 1 + 19.6T + 1.68e4T^{2} \)
11 \( 1 + 260.T + 1.61e5T^{2} \)
13 \( 1 + 65.1iT - 3.71e5T^{2} \)
17 \( 1 + 446. iT - 1.41e6T^{2} \)
19 \( 1 - 1.82e3T + 2.47e6T^{2} \)
23 \( 1 + 3.89e3T + 6.43e6T^{2} \)
29 \( 1 + 6.76e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.35e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.48e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.42e3iT - 1.15e8T^{2} \)
43 \( 1 - 159. iT - 1.47e8T^{2} \)
47 \( 1 - 4.46e3T + 2.29e8T^{2} \)
53 \( 1 - 1.44e4iT - 4.18e8T^{2} \)
61 \( 1 + 3.08e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.01iT - 1.35e9T^{2} \)
71 \( 1 - 5.91e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.62e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.61e4T + 3.07e9T^{2} \)
83 \( 1 - 5.18e4T + 3.93e9T^{2} \)
89 \( 1 - 7.34e4T + 5.58e9T^{2} \)
97 \( 1 + 1.43e5iT - 8.58e9T^{2} \)
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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

−10.07802817645178136705213231680, −9.276266622900307030035330048950, −8.170676469240707756628813273952, −7.62065725756900448119350070265, −6.29391029963888163574728261120, −5.25959185732173535892349909327, −4.20701039355993424546696054325, −3.01198529123993300392582474538, −1.80690939262145118020270808892, −0.59041729236039283107582927353, 2.01937905224040291093525533651, 3.03846396959091140468603109112, 3.71217684921401607587357181860, 4.99904583259637582531865003108, 6.14199373685426405103046086975, 7.28956504659679862028909536845, 7.977115438138796201301051269403, 9.350307843630533879625429116993, 10.31861825611809847245748410712, 10.80535650703987910425379732895

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