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Properties

Label 2-680-8.5-c1-0-58
Degree $2$
Conductor $680$
Sign $-0.481 - 0.876i$
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  + (−0.901 − 1.08i)2-s − 2.07i·3-s + (−0.375 + 1.96i)4-s − i·5-s + (−2.25 + 1.86i)6-s − 2.82·7-s + (2.47 − 1.36i)8-s − 1.29·9-s + (−1.08 + 0.901i)10-s − 4.37i·11-s + (4.07 + 0.778i)12-s − 4.97i·13-s + (2.54 + 3.08i)14-s − 2.07·15-s + (−3.71 − 1.47i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)2-s − 1.19i·3-s + (−0.187 + 0.982i)4-s − 0.447i·5-s + (−0.922 + 0.762i)6-s − 1.06·7-s + (0.876 − 0.481i)8-s − 0.432·9-s + (−0.344 + 0.284i)10-s − 1.31i·11-s + (1.17 + 0.224i)12-s − 1.38i·13-s + (0.681 + 0.823i)14-s − 0.535·15-s + (−0.929 − 0.368i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.481 - 0.876i$
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: \((2,\ 680,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274489 + 0.463836i\)
\(L(\frac12)\) \(\approx\) \(0.274489 + 0.463836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.901 + 1.08i)T \)
5 \( 1 + iT \)
17 \( 1 - T \)
good3 \( 1 + 2.07iT - 3T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 4.37iT - 11T^{2} \)
13 \( 1 + 4.97iT - 13T^{2} \)
19 \( 1 - 7.22iT - 19T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 - 2.86iT - 29T^{2} \)
31 \( 1 + 7.20T + 31T^{2} \)
37 \( 1 - 1.36iT - 37T^{2} \)
41 \( 1 - 4.80T + 41T^{2} \)
43 \( 1 - 8.40iT - 43T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 + 4.12iT - 53T^{2} \)
59 \( 1 - 8.49iT - 59T^{2} \)
61 \( 1 + 7.22iT - 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 + 3.06T + 71T^{2} \)
73 \( 1 - 2.56T + 73T^{2} \)
79 \( 1 + 8.71T + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 - 3.19T + 89T^{2} \)
97 \( 1 - 7.70T + 97T^{2} \)
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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

−9.989826015429095657383526322290, −9.035353826751883553661505353742, −7.999856825732757984940061001449, −7.74664842940339638464191232951, −6.39448474294225404492311403597, −5.65762825230352731883464618225, −3.76570782533290654380500351778, −2.97184980377345357824853771628, −1.52572628647373547822898472321, −0.35577456667711224744787742730, 2.24010930073344326697571797586, 3.93300839248660169958644400468, 4.63950776036859837399693915205, 5.79555472512902202960992879106, 6.92555653749277516567657709342, 7.24903027745947371422521943051, 8.854053658136656167461058605594, 9.480079015860117405256163831959, 9.862937110542449920157362435949, 10.66986970307724309772986735859

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