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Properties

Label 2-680-85.64-c1-0-13
Degree $2$
Conductor $680$
Sign $0.628 - 0.777i$
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  + (−0.803 + 0.803i)3-s + (2.02 + 0.938i)5-s + (0.286 + 0.286i)7-s + 1.70i·9-s + (4.32 − 4.32i)11-s − 2.18i·13-s + (−2.38 + 0.876i)15-s + (1.66 + 3.77i)17-s + 2.33i·19-s − 0.460·21-s + (−3.22 − 3.22i)23-s + (3.23 + 3.81i)25-s + (−3.78 − 3.78i)27-s + (5.33 + 5.33i)29-s + (1.07 + 1.07i)31-s + ⋯
L(s)  = 1  + (−0.463 + 0.463i)3-s + (0.907 + 0.419i)5-s + (0.108 + 0.108i)7-s + 0.569i·9-s + (1.30 − 1.30i)11-s − 0.605i·13-s + (−0.615 + 0.226i)15-s + (0.404 + 0.914i)17-s + 0.535i·19-s − 0.100·21-s + (−0.672 − 0.672i)23-s + (0.647 + 0.762i)25-s + (−0.728 − 0.728i)27-s + (0.990 + 0.990i)29-s + (0.193 + 0.193i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46424 + 0.699301i\)
\(L(\frac12)\) \(\approx\) \(1.46424 + 0.699301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.02 - 0.938i)T \)
17 \( 1 + (-1.66 - 3.77i)T \)
good3 \( 1 + (0.803 - 0.803i)T - 3iT^{2} \)
7 \( 1 + (-0.286 - 0.286i)T + 7iT^{2} \)
11 \( 1 + (-4.32 + 4.32i)T - 11iT^{2} \)
13 \( 1 + 2.18iT - 13T^{2} \)
19 \( 1 - 2.33iT - 19T^{2} \)
23 \( 1 + (3.22 + 3.22i)T + 23iT^{2} \)
29 \( 1 + (-5.33 - 5.33i)T + 29iT^{2} \)
31 \( 1 + (-1.07 - 1.07i)T + 31iT^{2} \)
37 \( 1 + (5.44 - 5.44i)T - 37iT^{2} \)
41 \( 1 + (-0.0474 + 0.0474i)T - 41iT^{2} \)
43 \( 1 - 3.53T + 43T^{2} \)
47 \( 1 - 2.96iT - 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 - 2.07iT - 59T^{2} \)
61 \( 1 + (0.0921 - 0.0921i)T - 61iT^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 + (4.26 + 4.26i)T + 71iT^{2} \)
73 \( 1 + (1.58 - 1.58i)T - 73iT^{2} \)
79 \( 1 + (-0.537 + 0.537i)T - 79iT^{2} \)
83 \( 1 + 6.51T + 83T^{2} \)
89 \( 1 + 6.83T + 89T^{2} \)
97 \( 1 + (-10.6 + 10.6i)T - 97iT^{2} \)
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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

−10.34063277762025392029362099039, −10.19943770478796905274058569885, −8.828955439852668344504553342273, −8.251490497664706829023831346975, −6.82238511856137126733861921010, −5.96760466967760320717490306762, −5.43283982868927092670649108634, −4.10736279449254357892153439133, −2.98050890019706629877778401886, −1.45905554631598029906410534664, 1.09660989789666197140711656158, 2.19808860689368871121100625222, 3.96640358842857514525566866364, 4.93371026183641606668578283493, 6.04206948550748470656628365961, 6.73252110630167856352362338857, 7.47533823529815197737903551665, 8.979129388338107393177174511203, 9.444936677912154923098042314108, 10.15072120676709989445708036092

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