Properties

Label 2-4950-5.4-c1-0-62
Degree $2$
Conductor $4950$
Sign $-0.894 + 0.447i$
Analytic cond. $39.5259$
Root an. cond. $6.28696$
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 + 11-s − 4i·13-s − 2·14-s + 16-s − 6i·17-s + 4·19-s i·22-s − 6i·23-s − 4·26-s + 2i·28-s + 6·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.755i·7-s + 0.353i·8-s + 0.301·11-s − 1.10i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.917·19-s − 0.213i·22-s − 1.25i·23-s − 0.784·26-s + 0.377i·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4950\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(39.5259\)
Root analytic conductor: \(6.28696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4950} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4950,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.746339886\)
\(L(\frac12)\) \(\approx\) \(1.746339886\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + 2iT - 7T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.238414568331591139577640142010, −7.16686295938038070670790008547, −6.66337877143013355261865359864, −5.58318888448732536257409824816, −4.79320397724313851884403347112, −4.26833418064369118492580917684, −2.98717809726799514704443415120, −2.85540869092853404967424156546, −1.23962517798520136727570334496, −0.55489720668283507282439958680, 1.25051033714618332800399908691, 2.25654632263624138682425558260, 3.44821593090815095482383272144, 4.16205507094380973754079920528, 5.02942439977974422593969569530, 5.80604571111637262822386563681, 6.32538742300598588344515614945, 7.10209218398243938230348323668, 7.78924244967713017272742860667, 8.664782909518632297295912105324

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