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Properties

Label 2-663-17.4-c1-0-14
Degree $2$
Conductor $663$
Sign $0.744 - 0.667i$
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  + 2.52i·2-s + (−0.707 + 0.707i)3-s − 4.39·4-s + (0.699 − 0.699i)5-s + (−1.78 − 1.78i)6-s + (−3.42 − 3.42i)7-s − 6.05i·8-s − 1.00i·9-s + (1.76 + 1.76i)10-s + (1.44 + 1.44i)11-s + (3.10 − 3.10i)12-s − 13-s + (8.65 − 8.65i)14-s + 0.989i·15-s + 6.52·16-s + (3.10 + 2.71i)17-s + ⋯
L(s)  = 1  + 1.78i·2-s + (−0.408 + 0.408i)3-s − 2.19·4-s + (0.312 − 0.312i)5-s + (−0.730 − 0.730i)6-s + (−1.29 − 1.29i)7-s − 2.14i·8-s − 0.333i·9-s + (0.559 + 0.559i)10-s + (0.436 + 0.436i)11-s + (0.897 − 0.897i)12-s − 0.277·13-s + (2.31 − 2.31i)14-s + 0.255i·15-s + 1.63·16-s + (0.753 + 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(5.29408\)
Root analytic conductor: \(2.30088\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742038 + 0.283745i\)
\(L(\frac12)\) \(\approx\) \(0.742038 + 0.283745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.10 - 2.71i)T \)
good2 \( 1 - 2.52iT - 2T^{2} \)
5 \( 1 + (-0.699 + 0.699i)T - 5iT^{2} \)
7 \( 1 + (3.42 + 3.42i)T + 7iT^{2} \)
11 \( 1 + (-1.44 - 1.44i)T + 11iT^{2} \)
19 \( 1 + 7.20iT - 19T^{2} \)
23 \( 1 + (-2.84 - 2.84i)T + 23iT^{2} \)
29 \( 1 + (-4.18 + 4.18i)T - 29iT^{2} \)
31 \( 1 + (-2.72 + 2.72i)T - 31iT^{2} \)
37 \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \)
41 \( 1 + (6.92 + 6.92i)T + 41iT^{2} \)
43 \( 1 + 4.44iT - 43T^{2} \)
47 \( 1 + 7.77T + 47T^{2} \)
53 \( 1 + 9.50iT - 53T^{2} \)
59 \( 1 + 12.2iT - 59T^{2} \)
61 \( 1 + (1.09 + 1.09i)T + 61iT^{2} \)
67 \( 1 + 0.675T + 67T^{2} \)
71 \( 1 + (-0.691 + 0.691i)T - 71iT^{2} \)
73 \( 1 + (7.77 - 7.77i)T - 73iT^{2} \)
79 \( 1 + (8.34 + 8.34i)T + 79iT^{2} \)
83 \( 1 - 4.89iT - 83T^{2} \)
89 \( 1 - 7.00T + 89T^{2} \)
97 \( 1 + (-2.67 + 2.67i)T - 97iT^{2} \)
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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

−10.09965484757711775815842879248, −9.633624085668590801491153118213, −8.906446166957766189060012683808, −7.64157235499280267086525531294, −6.92591496991983826069706745655, −6.36906068956210263540657420541, −5.35254883089090253468395499641, −4.47176062473792404961433073614, −3.54842474618715981176560518814, −0.50925481808433502692026320311, 1.29643953500063539466411339652, 2.77038915424162707082718344917, 3.14573186322945372383426784467, 4.70096327742457559898117715663, 5.86635287611093316448555970419, 6.55556994916152536373156189088, 8.236133225212153564018105771882, 9.057533883208598815576982917436, 10.00072069571128204475751391735, 10.23382952454665539427932421090

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