Properties

Label 2-280-280.237-c1-0-37
Degree $2$
Conductor $280$
Sign $0.993 + 0.117i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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)2-s + (2.42 − 2.42i)3-s + 2i·4-s + (−1.75 − 1.39i)5-s + 4.84·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 8.74i·9-s + (−0.359 − 3.14i)10-s + (4.84 + 4.84i)12-s + (−2.42 + 2.42i)13-s + 3.74i·14-s + (−7.61 + 0.870i)15-s − 4·16-s + (8.74 − 8.74i)18-s + 4.21i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (1.39 − 1.39i)3-s + i·4-s + (−0.782 − 0.622i)5-s + 1.97·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 2.91i·9-s + (−0.113 − 0.993i)10-s + (1.39 + 1.39i)12-s + (−0.672 + 0.672i)13-s + 0.999i·14-s + (−1.96 + 0.224i)15-s − 16-s + (2.06 − 2.06i)18-s + 0.968i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.993 + 0.117i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43361 - 0.143657i\)
\(L(\frac12)\) \(\approx\) \(2.43361 - 0.143657i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (1.75 + 1.39i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good3 \( 1 + (-2.42 + 2.42i)T - 3iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (2.42 - 2.42i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 4.21iT - 19T^{2} \)
23 \( 1 + (-0.741 + 0.741i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 0.625iT - 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 7.22T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 15.7iT - 79T^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−12.14794406954378740469750074070, −11.70662474739020182850190586613, −9.219391919088838554355068134792, −8.580659276223495287429676220682, −7.84035246896200280733236761687, −7.24686236270078508543369096429, −6.03987452824794498748629078326, −4.54041860947147154718760251169, −3.29633260994323964361219565220, −1.95294742844658589481979869451, 2.50078484901365260922877867185, 3.43386252445853009353079482271, 4.34093375767852753195323479610, 5.08513957121305046705508251564, 7.23089473952283514547462027141, 8.155719404591099884764360641186, 9.276969049380823432237973426935, 10.26632109788117262864929386418, 10.76527228105043320103136378779, 11.56215376497432947204533684274

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