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Properties

Label 2-680-17.13-c1-0-5
Degree $2$
Conductor $680$
Sign $0.476 - 0.879i$
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  + (−0.529 − 0.529i)3-s + (0.707 + 0.707i)5-s + (−3.59 + 3.59i)7-s − 2.43i·9-s + (2.90 − 2.90i)11-s + 4.66·13-s − 0.748i·15-s + (−2.39 + 3.35i)17-s + 7.16i·19-s + 3.80·21-s + (−3.15 + 3.15i)23-s + 1.00i·25-s + (−2.87 + 2.87i)27-s + (1.87 + 1.87i)29-s + (6.35 + 6.35i)31-s + ⋯
L(s)  = 1  + (−0.305 − 0.305i)3-s + (0.316 + 0.316i)5-s + (−1.35 + 1.35i)7-s − 0.813i·9-s + (0.876 − 0.876i)11-s + 1.29·13-s − 0.193i·15-s + (−0.580 + 0.814i)17-s + 1.64i·19-s + 0.830·21-s + (−0.658 + 0.658i)23-s + 0.200i·25-s + (−0.554 + 0.554i)27-s + (0.347 + 0.347i)29-s + (1.14 + 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.476 - 0.879i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01186 + 0.602706i\)
\(L(\frac12)\) \(\approx\) \(1.01186 + 0.602706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (2.39 - 3.35i)T \)
good3 \( 1 + (0.529 + 0.529i)T + 3iT^{2} \)
7 \( 1 + (3.59 - 3.59i)T - 7iT^{2} \)
11 \( 1 + (-2.90 + 2.90i)T - 11iT^{2} \)
13 \( 1 - 4.66T + 13T^{2} \)
19 \( 1 - 7.16iT - 19T^{2} \)
23 \( 1 + (3.15 - 3.15i)T - 23iT^{2} \)
29 \( 1 + (-1.87 - 1.87i)T + 29iT^{2} \)
31 \( 1 + (-6.35 - 6.35i)T + 31iT^{2} \)
37 \( 1 + (-4.97 - 4.97i)T + 37iT^{2} \)
41 \( 1 + (3.90 - 3.90i)T - 41iT^{2} \)
43 \( 1 - 0.399iT - 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 3.37iT - 53T^{2} \)
59 \( 1 + 2.33iT - 59T^{2} \)
61 \( 1 + (-1.28 + 1.28i)T - 61iT^{2} \)
67 \( 1 + 9.81T + 67T^{2} \)
71 \( 1 + (9.80 + 9.80i)T + 71iT^{2} \)
73 \( 1 + (-5.53 - 5.53i)T + 73iT^{2} \)
79 \( 1 + (-2.00 + 2.00i)T - 79iT^{2} \)
83 \( 1 + 2.06iT - 83T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + (-2.66 - 2.66i)T + 97iT^{2} \)
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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

−10.57747615206511169663975835471, −9.702129340369307273369792751825, −8.904809265068700301140788029361, −8.315217436111662420487311445988, −6.54410324970873246398937360928, −6.22613197271725708040248548999, −5.78483362282967354983385550294, −3.75558866940510259882290779877, −3.14411831299555936051689261386, −1.46792483886795350775024389551, 0.70282838294999846100221352286, 2.54063997490407563530621677737, 4.09091865264166081589067272850, 4.51773800298868329987807162317, 6.02325374383459762217515605139, 6.73668232632110549740704772073, 7.51681546524988981454481665024, 8.871998727747467176928401159713, 9.583347399912910146389616680265, 10.32116236392436773459211927260

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