Properties

Label 2-1638-13.10-c1-0-8
Degree $2$
Conductor $1638$
Sign $0.702 - 0.711i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 3.46i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (1.73 + 2.99i)10-s + (−3.46 + 2i)11-s + (2.59 + 2.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−3.23 + 5.59i)17-s + (6.46 + 3.73i)19-s + (−2.99 − 1.73i)20-s + (1.99 − 3.46i)22-s + (−2.86 − 4.96i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.54i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (0.547 + 0.948i)10-s + (−1.04 + 0.603i)11-s + (0.720 + 0.693i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.783 + 1.35i)17-s + (1.48 + 0.856i)19-s + (−0.670 − 0.387i)20-s + (0.426 − 0.738i)22-s + (−0.597 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9665751636\)
\(L(\frac12)\) \(\approx\) \(0.9665751636\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + (3.46 - 2i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.86 + 4.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4 - 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.46iT - 31T^{2} \)
37 \( 1 + (4.26 - 2.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.23 + 3.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.535iT - 47T^{2} \)
53 \( 1 - 8.26T + 53T^{2} \)
59 \( 1 + (-5.42 - 3.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.59 + 4.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.23 + 3.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.66 + 0.964i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 1.73iT - 83T^{2} \)
89 \( 1 + (-10.1 + 5.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.7 - 6.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.211594339767257658222668969018, −8.672870738027657939133332436236, −8.143034879533472605661205333117, −7.21134763022586973497923810552, −6.28635588150508714572636986654, −5.39114257964331780268303225129, −4.68506383341630348296675289737, −3.68145815522098628826153384502, −2.02416792366586652134035658531, −1.04673860130729398242749355376, 0.54466327255355096686042607235, 2.62433436135937043861215829117, 2.79002842510002650767909560168, 3.85702603251715300755257914005, 5.40041541601450494647431667926, 6.13767594877675914901427197252, 7.14787587067279238454697148857, 7.59372137587748929922739397065, 8.465869903066865333912391843401, 9.586814087298611172711039607537

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