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Properties

Label 2-680-85.4-c1-0-5
Degree $2$
Conductor $680$
Sign $0.989 + 0.143i$
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  + (−1.69 − 1.69i)3-s + (−2.23 + 0.0897i)5-s + (−2.68 + 2.68i)7-s + 2.76i·9-s + (1.62 + 1.62i)11-s − 4.79i·13-s + (3.94 + 3.64i)15-s + (2.73 + 3.08i)17-s − 0.473i·19-s + 9.13·21-s + (−0.747 + 0.747i)23-s + (4.98 − 0.401i)25-s + (−0.390 + 0.390i)27-s + (6.84 − 6.84i)29-s + (−3.38 + 3.38i)31-s + ⋯
L(s)  = 1  + (−0.980 − 0.980i)3-s + (−0.999 + 0.0401i)5-s + (−1.01 + 1.01i)7-s + 0.923i·9-s + (0.488 + 0.488i)11-s − 1.33i·13-s + (1.01 + 0.940i)15-s + (0.662 + 0.749i)17-s − 0.108i·19-s + 1.99·21-s + (−0.155 + 0.155i)23-s + (0.996 − 0.0802i)25-s + (−0.0752 + 0.0752i)27-s + (1.27 − 1.27i)29-s + (−0.608 + 0.608i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.989 + 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678383 - 0.0487747i\)
\(L(\frac12)\) \(\approx\) \(0.678383 - 0.0487747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0897i)T \)
17 \( 1 + (-2.73 - 3.08i)T \)
good3 \( 1 + (1.69 + 1.69i)T + 3iT^{2} \)
7 \( 1 + (2.68 - 2.68i)T - 7iT^{2} \)
11 \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \)
13 \( 1 + 4.79iT - 13T^{2} \)
19 \( 1 + 0.473iT - 19T^{2} \)
23 \( 1 + (0.747 - 0.747i)T - 23iT^{2} \)
29 \( 1 + (-6.84 + 6.84i)T - 29iT^{2} \)
31 \( 1 + (3.38 - 3.38i)T - 31iT^{2} \)
37 \( 1 + (-2.83 - 2.83i)T + 37iT^{2} \)
41 \( 1 + (-1.76 - 1.76i)T + 41iT^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 + 2.14T + 53T^{2} \)
59 \( 1 - 6.46iT - 59T^{2} \)
61 \( 1 + (0.354 + 0.354i)T + 61iT^{2} \)
67 \( 1 - 13.9iT - 67T^{2} \)
71 \( 1 + (-4.21 + 4.21i)T - 71iT^{2} \)
73 \( 1 + (-11.2 - 11.2i)T + 73iT^{2} \)
79 \( 1 + (-5.88 - 5.88i)T + 79iT^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + 9.11T + 89T^{2} \)
97 \( 1 + (-6.39 - 6.39i)T + 97iT^{2} \)
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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

−10.61341023934589680669859995642, −9.707521940323733886101504103684, −8.526025064811620322825542572757, −7.73789398238182406350880745748, −6.82867221082939533799572784440, −6.07502388669957063633781471159, −5.34140638886635607807580305114, −3.85349112852955236659447199075, −2.66895031337748473130350695188, −0.833078185244814787282807256012, 0.63520352479767384723378764849, 3.30409440732127778274038415656, 4.10025797370521962789821156740, 4.75537025855424757800102371517, 6.08198016093263804400369790501, 6.85955274411796326613088699935, 7.73824178836678072407279026089, 9.154793814933497539634483855631, 9.655649914290587770158453574584, 10.79923136084438691004354912378

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