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Properties

Label 2-578-17.13-c1-0-8
Degree $2$
Conductor $578$
Sign $0.788 - 0.615i$
Analytic cond. $4.61535$
Root an. cond. $2.14833$
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  + i·2-s − 4-s + (1 + i)5-s − i·8-s − 3i·9-s + (−1 + i)10-s + (4 − 4i)11-s + 4·13-s + 16-s + 3·18-s + 4i·19-s + (−1 − i)20-s + (4 + 4i)22-s + (−4 + 4i)23-s − 3i·25-s + 4i·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.447 + 0.447i)5-s − 0.353i·8-s − i·9-s + (−0.316 + 0.316i)10-s + (1.20 − 1.20i)11-s + 1.10·13-s + 0.250·16-s + 0.707·18-s + 0.917i·19-s + (−0.223 − 0.223i)20-s + (0.852 + 0.852i)22-s + (−0.834 + 0.834i)23-s − 0.600i·25-s + 0.784i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(578\)    =    \(2 \cdot 17^{2}\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(4.61535\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{578} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 578,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54297 + 0.531014i\)
\(L(\frac12)\) \(\approx\) \(1.54297 + 0.531014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 + 3iT^{2} \)
5 \( 1 + (-1 - i)T + 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (-4 + 4i)T - 11iT^{2} \)
13 \( 1 - 4T + 13T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (4 - 4i)T - 23iT^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 + (-4 - 4i)T + 31iT^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (1 - i)T - 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + (-9 + 9i)T - 61iT^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (-4 - 4i)T + 71iT^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + (8 - 8i)T - 79iT^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
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

−10.70518616506848593179418024427, −9.799632232840517976514485415321, −8.866503520279624932003616231414, −8.338513064455486644488038066998, −6.96399608342570129445661753787, −6.18017623178781909229671196896, −5.79206651811564046971533723834, −4.02913392557112067536592768971, −3.33343509275852209901860721947, −1.21675379765085187782761711890, 1.37783501951696993644670835664, 2.43616500311287714763409733791, 4.05693569417186939224171196619, 4.75970827830507497575162266537, 5.94870709376521333908882557000, 7.02619771862001224052744534262, 8.251319101344992412473892727932, 9.004978716998814113332348355993, 9.840985070138082614710686310899, 10.57613690517293325088691929115

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