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Properties

Label 2-663-17.13-c1-0-28
Degree $2$
Conductor $663$
Sign $-0.749 + 0.661i$
Analytic cond. $5.29408$
Root an. cond. $2.30088$
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  − 1.49i·2-s + (−0.707 − 0.707i)3-s − 0.225·4-s + (0.731 + 0.731i)5-s + (−1.05 + 1.05i)6-s + (−0.0312 + 0.0312i)7-s − 2.64i·8-s + 1.00i·9-s + (1.09 − 1.09i)10-s + (2.62 − 2.62i)11-s + (0.159 + 0.159i)12-s − 13-s + (0.0465 + 0.0465i)14-s − 1.03i·15-s − 4.39·16-s + (2.32 + 3.40i)17-s + ⋯
L(s)  = 1  − 1.05i·2-s + (−0.408 − 0.408i)3-s − 0.112·4-s + (0.327 + 0.327i)5-s + (−0.430 + 0.430i)6-s + (−0.0118 + 0.0118i)7-s − 0.936i·8-s + 0.333i·9-s + (0.345 − 0.345i)10-s + (0.792 − 0.792i)11-s + (0.0459 + 0.0459i)12-s − 0.277·13-s + (0.0124 + 0.0124i)14-s − 0.267i·15-s − 1.09·16-s + (0.565 + 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530722 - 1.40368i\)
\(L(\frac12)\) \(\approx\) \(0.530722 - 1.40368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + T \)
17 \( 1 + (-2.32 - 3.40i)T \)
good2 \( 1 + 1.49iT - 2T^{2} \)
5 \( 1 + (-0.731 - 0.731i)T + 5iT^{2} \)
7 \( 1 + (0.0312 - 0.0312i)T - 7iT^{2} \)
11 \( 1 + (-2.62 + 2.62i)T - 11iT^{2} \)
19 \( 1 + 4.41iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + (1.50 + 1.50i)T + 29iT^{2} \)
31 \( 1 + (-0.427 - 0.427i)T + 31iT^{2} \)
37 \( 1 + (7.53 + 7.53i)T + 37iT^{2} \)
41 \( 1 + (-1.87 + 1.87i)T - 41iT^{2} \)
43 \( 1 - 3.13iT - 43T^{2} \)
47 \( 1 - 1.35T + 47T^{2} \)
53 \( 1 - 2.47iT - 53T^{2} \)
59 \( 1 - 6.41iT - 59T^{2} \)
61 \( 1 + (7.89 - 7.89i)T - 61iT^{2} \)
67 \( 1 + 0.572T + 67T^{2} \)
71 \( 1 + (-5.12 - 5.12i)T + 71iT^{2} \)
73 \( 1 + (-1.11 - 1.11i)T + 73iT^{2} \)
79 \( 1 + (-11.9 + 11.9i)T - 79iT^{2} \)
83 \( 1 - 5.05iT - 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (8.89 + 8.89i)T + 97iT^{2} \)
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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

−10.69894519820782493576930498216, −9.526594919865017270262926971754, −8.708571473578721284784435623618, −7.39605864828036369718391826546, −6.55490599939288702541674078245, −5.82269356580946288722353106198, −4.37073024216121039773512768775, −3.22559597377655606154335475957, −2.18761481008388289920941809275, −0.891325117775065735013591077521, 1.71923708907459251071549680258, 3.47296722220154361447396974232, 4.91261076951850650906724823744, 5.39503210796783902581342971466, 6.47878296823147438814232322214, 7.16456003956765738924711885187, 8.063921878053111962078228649205, 9.192456174368227210305713221761, 9.745455394372043777193308056158, 10.81330662974776225360606532313

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