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Properties

Label 2-354-59.21-c1-0-8
Degree $2$
Conductor $354$
Sign $0.981 - 0.191i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.856 + 0.515i)2-s + (0.370 − 0.928i)3-s + (0.468 + 0.883i)4-s + (3.29 + 0.358i)5-s + (0.796 − 0.605i)6-s + (−0.183 − 0.0617i)7-s + (−0.0541 + 0.998i)8-s + (−0.725 − 0.687i)9-s + (2.64 + 2.00i)10-s + (−0.940 − 1.38i)11-s + (0.994 − 0.108i)12-s + (−1.67 + 1.58i)13-s + (−0.125 − 0.147i)14-s + (1.55 − 2.93i)15-s + (−0.561 + 0.827i)16-s + (−1.50 + 0.507i)17-s + ⋯
L(s)  = 1  + (0.605 + 0.364i)2-s + (0.213 − 0.536i)3-s + (0.234 + 0.441i)4-s + (1.47 + 0.160i)5-s + (0.325 − 0.247i)6-s + (−0.0693 − 0.0233i)7-s + (−0.0191 + 0.353i)8-s + (−0.241 − 0.229i)9-s + (0.835 + 0.634i)10-s + (−0.283 − 0.418i)11-s + (0.286 − 0.0312i)12-s + (−0.465 + 0.440i)13-s + (−0.0334 − 0.0394i)14-s + (0.401 − 0.756i)15-s + (−0.140 + 0.206i)16-s + (−0.365 + 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.981 - 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30329 + 0.223040i\)
\(L(\frac12)\) \(\approx\) \(2.30329 + 0.223040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.856 - 0.515i)T \)
3 \( 1 + (-0.370 + 0.928i)T \)
59 \( 1 + (7.34 + 2.24i)T \)
good5 \( 1 + (-3.29 - 0.358i)T + (4.88 + 1.07i)T^{2} \)
7 \( 1 + (0.183 + 0.0617i)T + (5.57 + 4.23i)T^{2} \)
11 \( 1 + (0.940 + 1.38i)T + (-4.07 + 10.2i)T^{2} \)
13 \( 1 + (1.67 - 1.58i)T + (0.703 - 12.9i)T^{2} \)
17 \( 1 + (1.50 - 0.507i)T + (13.5 - 10.2i)T^{2} \)
19 \( 1 + (-0.143 + 0.875i)T + (-18.0 - 6.06i)T^{2} \)
23 \( 1 + (-0.528 + 1.90i)T + (-19.7 - 11.8i)T^{2} \)
29 \( 1 + (1.94 - 1.16i)T + (13.5 - 25.6i)T^{2} \)
31 \( 1 + (0.0829 + 0.506i)T + (-29.3 + 9.89i)T^{2} \)
37 \( 1 + (-0.111 - 2.05i)T + (-36.7 + 4.00i)T^{2} \)
41 \( 1 + (-2.31 - 8.33i)T + (-35.1 + 21.1i)T^{2} \)
43 \( 1 + (-7.12 + 10.5i)T + (-15.9 - 39.9i)T^{2} \)
47 \( 1 + (5.54 - 0.603i)T + (45.9 - 10.1i)T^{2} \)
53 \( 1 + (7.64 - 5.81i)T + (14.1 - 51.0i)T^{2} \)
61 \( 1 + (-2.82 - 1.69i)T + (28.5 + 53.8i)T^{2} \)
67 \( 1 + (-0.0874 + 1.61i)T + (-66.6 - 7.24i)T^{2} \)
71 \( 1 + (12.6 - 1.37i)T + (69.3 - 15.2i)T^{2} \)
73 \( 1 + (2.70 + 3.18i)T + (-11.8 + 72.0i)T^{2} \)
79 \( 1 + (-2.00 - 5.04i)T + (-57.3 + 54.3i)T^{2} \)
83 \( 1 + (5.97 + 2.76i)T + (53.7 + 63.2i)T^{2} \)
89 \( 1 + (-1.81 + 1.08i)T + (41.6 - 78.6i)T^{2} \)
97 \( 1 + (-9.22 + 10.8i)T + (-15.6 - 95.7i)T^{2} \)
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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

−11.64262468586751329528315292179, −10.60620445933959956829414667732, −9.563820036600503624240710113921, −8.697372702325435558379300483697, −7.48043576214425680693140326099, −6.49670567959276108108045797468, −5.84377941192648267505101938056, −4.72075647632331489282226148004, −3.01378221061855006447109482140, −1.93085761874285235005360025416, 1.91135209679078042219011226611, 3.00770884144081280778063381866, 4.53514100892767371147498594377, 5.42519282344197156844213220591, 6.24848915252359769145587461039, 7.61208736500022380218195382024, 9.087247185654730168825513089360, 9.733523183108313911059830451951, 10.40691338928299400584810968257, 11.33847871146323753625641242952

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