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Properties

Label 2-680-8.5-c1-0-17
Degree $2$
Conductor $680$
Sign $-0.999 - 0.0128i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
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  + (1.22 + 0.712i)2-s + 2.79i·3-s + (0.985 + 1.74i)4-s + i·5-s + (−1.98 + 3.40i)6-s − 1.58·7-s + (−0.0363 + 2.82i)8-s − 4.78·9-s + (−0.712 + 1.22i)10-s − 3.33i·11-s + (−4.85 + 2.74i)12-s + 0.914i·13-s + (−1.93 − 1.12i)14-s − 2.79·15-s + (−2.05 + 3.42i)16-s + 17-s + ⋯
L(s)  = 1  + (0.863 + 0.503i)2-s + 1.61i·3-s + (0.492 + 0.870i)4-s + 0.447i·5-s + (−0.811 + 1.39i)6-s − 0.597·7-s + (−0.0128 + 0.999i)8-s − 1.59·9-s + (−0.225 + 0.386i)10-s − 1.00i·11-s + (−1.40 + 0.793i)12-s + 0.253i·13-s + (−0.516 − 0.300i)14-s − 0.720·15-s + (−0.514 + 0.857i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.999 - 0.0128i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.999 - 0.0128i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0140375 + 2.18199i\)
\(L(\frac12)\) \(\approx\) \(0.0140375 + 2.18199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.712i)T \)
5 \( 1 - iT \)
17 \( 1 - T \)
good3 \( 1 - 2.79iT - 3T^{2} \)
7 \( 1 + 1.58T + 7T^{2} \)
11 \( 1 + 3.33iT - 11T^{2} \)
13 \( 1 - 0.914iT - 13T^{2} \)
19 \( 1 + 1.61iT - 19T^{2} \)
23 \( 1 - 8.26T + 23T^{2} \)
29 \( 1 - 6.54iT - 29T^{2} \)
31 \( 1 - 4.97T + 31T^{2} \)
37 \( 1 - 5.16iT - 37T^{2} \)
41 \( 1 + 2.41T + 41T^{2} \)
43 \( 1 + 7.62iT - 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 - 9.94iT - 53T^{2} \)
59 \( 1 + 3.83iT - 59T^{2} \)
61 \( 1 + 7.39iT - 61T^{2} \)
67 \( 1 - 4.67iT - 67T^{2} \)
71 \( 1 + 3.26T + 71T^{2} \)
73 \( 1 + 2.86T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 8.54iT - 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−10.95742523634637432205797413398, −10.20351410960623273464051091668, −9.116908212681594063833654182280, −8.521986647643027130092845012857, −7.15002423509857127068750897170, −6.27255808663630512005605365053, −5.31288985446562537586019663760, −4.56002185617176533653489238036, −3.36820186153873599808193506636, −3.03608914153083791252071731667, 0.900288591771564862948542488428, 2.06453265211296190237578198586, 3.09560346871019338294103705415, 4.53075968240824113166771037971, 5.61646603283875796388102319007, 6.50206425106367514224503930399, 7.16108817382501237164507109507, 8.060796101370843525621200376053, 9.327451441916024593866166716544, 10.14624056730959125962979896453

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