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Properties

Label 2-4950-5.4-c1-0-44
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 − i·8-s − 11-s − 2i·13-s + 16-s + 2i·17-s − 8·19-s − i·22-s + 4i·23-s + 2·26-s + 2·29-s + 8·31-s + i·32-s − 2·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.353i·8-s − 0.301·11-s − 0.554i·13-s + 0.250·16-s + 0.485i·17-s − 1.83·19-s − 0.213i·22-s + 0.834i·23-s + 0.392·26-s + 0.371·29-s + 1.43·31-s + 0.176i·32-s − 0.342·34-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

\(L(1)\) \(\approx\) \(1.141302164\)
\(L(\frac12)\) \(\approx\) \(1.141302164\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
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

−8.274144457258819576838981695341, −7.49213978741928378738243784514, −6.70286002135309079424865634465, −6.09856681936011561417592181482, −5.39639369471083816992935942203, −4.56636531017642371479162862705, −3.86680968674448523746282585281, −2.86566000674108629309979616205, −1.79204601451514659651867796755, −0.37034063804079065692574551207, 0.918663931393695322648898048333, 2.16786768706753161051850350886, 2.71135756387754777063831118542, 3.83126649431812667738958271080, 4.52363379722262365840507539788, 5.11862830360734451752871630259, 6.30226464008853465133974708387, 6.67415607435903460062451189897, 7.77007828511563878924044405359, 8.515332137148589082014493419698

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