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Properties

Label 2-680-85.4-c1-0-4
Degree $2$
Conductor $680$
Sign $0.197 - 0.980i$
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 − i)3-s + (1 + 2i)5-s + (−1 + i)7-s − i·9-s + (−1 − i)11-s + 4i·13-s + (1 − 3i)15-s + (1 + 4i)17-s + 2i·19-s + 2·21-s + (−1 + i)23-s + (−3 + 4i)25-s + (−4 + 4i)27-s + (−1 + i)29-s + (3 − 3i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.447 + 0.894i)5-s + (−0.377 + 0.377i)7-s − 0.333i·9-s + (−0.301 − 0.301i)11-s + 1.10i·13-s + (0.258 − 0.774i)15-s + (0.242 + 0.970i)17-s + 0.458i·19-s + 0.436·21-s + (−0.208 + 0.208i)23-s + (−0.600 + 0.800i)25-s + (−0.769 + 0.769i)27-s + (−0.185 + 0.185i)29-s + (0.538 − 0.538i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.197 - 0.980i$
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: \((2,\ 680,\ (\ :1/2),\ 0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762991 + 0.624314i\)
\(L(\frac12)\) \(\approx\) \(0.762991 + 0.624314i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2i)T \)
17 \( 1 + (-1 - 4i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (1 - i)T - 29iT^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + (-7 - 7i)T + 61iT^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + (-11 + 11i)T - 71iT^{2} \)
73 \( 1 + (7 + 7i)T + 73iT^{2} \)
79 \( 1 + (7 + 7i)T + 79iT^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−10.75227346075054668186379534739, −9.856052856847112327517801967988, −9.099613978411184334948634543238, −7.904237940730556258796586112853, −6.92452959911594060066744817243, −6.19066820880002306896980718421, −5.72869083098283580177740144764, −4.08735658591105700612540569845, −2.90988881439706388512347512292, −1.60217272815744785465613517121, 0.56679093429933113851747296220, 2.42767364814361590526170094616, 3.91907614118312828060668416747, 5.12650900895797829435051017096, 5.34724257173719048720579929272, 6.65429512895189470741034803590, 7.77190327029488176361995617033, 8.610676559349124628417149834915, 9.840002603775836822458373304433, 10.04522248842947136815887678194

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