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Properties

Label 2-4050-5.4-c1-0-28
Degree $2$
Conductor $4050$
Sign $-0.447 - 0.894i$
Analytic cond. $32.3394$
Root an. cond. $5.68677$
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 + 2i·7-s − i·8-s + 4i·13-s − 2·14-s + 16-s − 6i·17-s + 7·19-s − 4·26-s − 2i·28-s + 6·29-s − 10·31-s + i·32-s + 6·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.755i·7-s − 0.353i·8-s + 1.10i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s + 1.60·19-s − 0.784·26-s − 0.377i·28-s + 1.11·29-s − 1.79·31-s + 0.176i·32-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4050\)    =    \(2 \cdot 3^{4} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(32.3394\)
Root analytic conductor: \(5.68677\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4050,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692424725\)
\(L(\frac12)\) \(\approx\) \(1.692424725\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 9T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 - 17iT - 97T^{2} \)
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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

−8.805856177567859002070517205193, −7.69902753231507802638820379407, −7.31226451940302336730862685322, −6.50318940822930695702618271278, −5.70264484843409895762071121840, −5.07810895614880171917144638683, −4.34608489938324046090043032395, −3.25571159188896679213064215776, −2.38779048958888832617022952129, −1.04347743841135907048745303464, 0.59433246210078581894413624233, 1.53438325158638567969320821890, 2.70865202978629777466092573965, 3.60672087724269147412590314271, 4.09099822425226783812750964769, 5.28887201959665023928439923614, 5.69807111918314342837917991461, 6.86909677353774157660711830655, 7.57722258358808735598727363065, 8.238351007039741974527337009330

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