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Properties

Label 2-1638-7.4-c1-0-21
Degree $2$
Conductor $1638$
Sign $0.851 + 0.524i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.90 + 3.29i)5-s + (−2.64 + 0.0932i)7-s + 0.999·8-s + (1.90 − 3.29i)10-s + (2.64 − 4.57i)11-s + 13-s + (1.40 + 2.24i)14-s + (−0.5 − 0.866i)16-s + (3.07 − 5.33i)17-s + (−2.74 − 4.74i)19-s − 3.80·20-s − 5.28·22-s + (3.14 + 5.44i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.850 + 1.47i)5-s + (−0.999 + 0.0352i)7-s + 0.353·8-s + (0.601 − 1.04i)10-s + (0.797 − 1.38i)11-s + 0.277·13-s + (0.374 + 0.599i)14-s + (−0.125 − 0.216i)16-s + (0.746 − 1.29i)17-s + (−0.628 − 1.08i)19-s − 0.850·20-s − 1.12·22-s + (0.655 + 1.13i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532862136\)
\(L(\frac12)\) \(\approx\) \(1.532862136\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (2.64 - 0.0932i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.64 + 4.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.07 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.74 + 4.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-4.24 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.661 + 1.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.32T + 41T^{2} \)
43 \( 1 - 1.61T + 43T^{2} \)
47 \( 1 + (-3.54 - 6.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.177 - 0.306i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.838 - 1.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.88 - 8.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.22 + 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (0.855 - 1.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.43 + 2.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.44T + 83T^{2} \)
89 \( 1 + (-5.33 - 9.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.15T + 97T^{2} \)
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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

−9.456362158182323713104358527801, −8.901781828356458379530008253120, −7.65819702451645617559272060081, −6.75829759737418791522997472093, −6.29961448490895540964504427622, −5.38946640509889204010798547266, −3.78520640185651966804091855623, −3.04875502092825324714609637186, −2.51754293581996703154127976994, −0.842712180743652682227729762331, 1.07008239557736581730294705623, 1.98323635552512095912782838912, 3.76113359847357179035975204617, 4.63728187483459234183586652583, 5.46069473182972903081592398541, 6.38265059534107051529470595859, 6.76547819381017869767065007958, 8.157788646928799727181102563903, 8.647942305422870169367234377015, 9.356262844663833669866758310120

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