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Properties

Label 2-680-85.64-c1-0-4
Degree $2$
Conductor $680$
Sign $0.0832 - 0.996i$
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  + (0.253 − 0.253i)3-s + (1.48 + 1.66i)5-s + (−3.26 − 3.26i)7-s + 2.87i·9-s + (−2.95 + 2.95i)11-s + 5.14i·13-s + (0.799 + 0.0451i)15-s + (4.07 + 0.613i)17-s + 1.32i·19-s − 1.65·21-s + (2.97 + 2.97i)23-s + (−0.563 + 4.96i)25-s + (1.48 + 1.48i)27-s + (−5.42 − 5.42i)29-s + (5.27 + 5.27i)31-s + ⋯
L(s)  = 1  + (0.146 − 0.146i)3-s + (0.666 + 0.745i)5-s + (−1.23 − 1.23i)7-s + 0.957i·9-s + (−0.890 + 0.890i)11-s + 1.42i·13-s + (0.206 + 0.0116i)15-s + (0.988 + 0.148i)17-s + 0.304i·19-s − 0.360·21-s + (0.620 + 0.620i)23-s + (−0.112 + 0.993i)25-s + (0.286 + 0.286i)27-s + (−1.00 − 1.00i)29-s + (0.947 + 0.947i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.0832 - 0.996i$
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: \((2,\ 680,\ (\ :1/2),\ 0.0832 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931844 + 0.857279i\)
\(L(\frac12)\) \(\approx\) \(0.931844 + 0.857279i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.66i)T \)
17 \( 1 + (-4.07 - 0.613i)T \)
good3 \( 1 + (-0.253 + 0.253i)T - 3iT^{2} \)
7 \( 1 + (3.26 + 3.26i)T + 7iT^{2} \)
11 \( 1 + (2.95 - 2.95i)T - 11iT^{2} \)
13 \( 1 - 5.14iT - 13T^{2} \)
19 \( 1 - 1.32iT - 19T^{2} \)
23 \( 1 + (-2.97 - 2.97i)T + 23iT^{2} \)
29 \( 1 + (5.42 + 5.42i)T + 29iT^{2} \)
31 \( 1 + (-5.27 - 5.27i)T + 31iT^{2} \)
37 \( 1 + (0.749 - 0.749i)T - 37iT^{2} \)
41 \( 1 + (-7.59 + 7.59i)T - 41iT^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + 4.44iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 9.62iT - 59T^{2} \)
61 \( 1 + (-4.29 + 4.29i)T - 61iT^{2} \)
67 \( 1 - 5.25iT - 67T^{2} \)
71 \( 1 + (-0.879 - 0.879i)T + 71iT^{2} \)
73 \( 1 + (-3.95 + 3.95i)T - 73iT^{2} \)
79 \( 1 + (-1.97 + 1.97i)T - 79iT^{2} \)
83 \( 1 + 2.75T + 83T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + (7.24 - 7.24i)T - 97iT^{2} \)
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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

−10.39583830505698589573054271980, −10.02629362326116835447028900822, −9.273084070519340627929338366138, −7.72198556560039057845592777043, −7.20897547010352352488468115175, −6.46536493785432933181678198616, −5.31315047953634490592805397097, −4.08675158560233065505252637578, −2.97390256506585476520319419181, −1.80703038507079874511794028000, 0.64824368234881242370024159107, 2.75749024312211438910610327510, 3.28004502973467760398334490661, 5.08499702775179471789909044012, 5.80910943969149862327597695248, 6.35460772103908041566266776972, 7.936781596609548710664338083559, 8.695766244551458500015343496167, 9.501486079336020504680390464445, 9.936743011997094985756651936893

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