Properties

Label 2-280-280.237-c1-0-34
Degree $2$
Conductor $280$
Sign $-0.177 + 0.984i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (0.649 − 1.25i)2-s + (0.639 − 0.639i)3-s + (−1.15 − 1.63i)4-s + (1.45 + 1.69i)5-s + (−0.388 − 1.21i)6-s + (0.0280 − 2.64i)7-s + (−2.80 + 0.395i)8-s + 2.18i·9-s + (3.07 − 0.728i)10-s − 5.47i·11-s + (−1.78 − 0.303i)12-s + (3.89 − 3.89i)13-s + (−3.30 − 1.75i)14-s + (2.01 + 0.153i)15-s + (−1.32 + 3.77i)16-s + (−3.30 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.459 − 0.888i)2-s + (0.369 − 0.369i)3-s + (−0.578 − 0.815i)4-s + (0.651 + 0.758i)5-s + (−0.158 − 0.497i)6-s + (0.0106 − 0.999i)7-s + (−0.990 + 0.139i)8-s + 0.727i·9-s + (0.973 − 0.230i)10-s − 1.65i·11-s + (−0.514 − 0.0875i)12-s + (1.07 − 1.07i)13-s + (−0.883 − 0.468i)14-s + (0.520 + 0.0396i)15-s + (−0.330 + 0.943i)16-s + (−0.801 + 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.177 + 0.984i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.177 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16458 - 1.39307i\)
\(L(\frac12)\) \(\approx\) \(1.16458 - 1.39307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.649 + 1.25i)T \)
5 \( 1 + (-1.45 - 1.69i)T \)
7 \( 1 + (-0.0280 + 2.64i)T \)
good3 \( 1 + (-0.639 + 0.639i)T - 3iT^{2} \)
11 \( 1 + 5.47iT - 11T^{2} \)
13 \( 1 + (-3.89 + 3.89i)T - 13iT^{2} \)
17 \( 1 + (3.30 - 3.30i)T - 17iT^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 + (0.137 - 0.137i)T - 23iT^{2} \)
29 \( 1 + 2.31T + 29T^{2} \)
31 \( 1 - 4.44iT - 31T^{2} \)
37 \( 1 + (-4.36 + 4.36i)T - 37iT^{2} \)
41 \( 1 - 4.27iT - 41T^{2} \)
43 \( 1 + (-1.77 - 1.77i)T + 43iT^{2} \)
47 \( 1 + (0.164 - 0.164i)T - 47iT^{2} \)
53 \( 1 + (-2.76 - 2.76i)T + 53iT^{2} \)
59 \( 1 + 2.55iT - 59T^{2} \)
61 \( 1 + 0.863T + 61T^{2} \)
67 \( 1 + (4.38 - 4.38i)T - 67iT^{2} \)
71 \( 1 + 3.84T + 71T^{2} \)
73 \( 1 + (-8.74 - 8.74i)T + 73iT^{2} \)
79 \( 1 + 14.5iT - 79T^{2} \)
83 \( 1 + (0.395 - 0.395i)T - 83iT^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \)
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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

−11.20651738522011306081347817903, −10.77149302121301855475010960156, −10.20880234895941803750620325077, −8.744097757557562005794031227010, −7.87949190333528147211809173680, −6.32699456163457323180624725968, −5.60516752643558245388187473020, −3.84481529374059995511557689597, −2.96604133755799636015749813785, −1.42592643414438673810186144819, 2.38741557311574729875579550350, 4.18147330470203280287188167003, 4.93360125769795746643151689005, 6.17186252945669838305015743227, 6.98708130723491300385820423204, 8.510781176001352294296849896687, 9.282707442865158002689129315042, 9.500565578215089528955780581244, 11.54687077460869283427444864656, 12.32559653912523112172837884329

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