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Properties

Label 2-4650-5.4-c1-0-68
Degree $2$
Conductor $4650$
Sign $-0.447 + 0.894i$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
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 + i·3-s − 4-s + 6-s − 3i·7-s + i·8-s − 9-s + 3·11-s − i·12-s + 2i·13-s − 3·14-s + 16-s − 4i·17-s + i·18-s + 3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.13i·7-s + 0.353i·8-s − 0.333·9-s + 0.904·11-s − 0.288i·12-s + 0.554i·13-s − 0.801·14-s + 0.250·16-s − 0.970i·17-s + 0.235i·18-s + 0.688·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 31\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(37.1304\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4650} (3349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.578750717\)
\(L(\frac12)\) \(\approx\) \(1.578750717\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
31 \( 1 - T \)
good7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.164878581881082805386353012400, −7.36327569423185384288533927900, −6.69828176887307990528225837183, −5.77891594023404932690400278135, −4.71459192377336990997630120274, −4.30247899995065091114816768902, −3.55666236049656402587146008709, −2.73378393516684869698336643202, −1.50807896490172012490261394488, −0.49163792157797104475249031551, 1.14428449258354657984277874959, 2.14087115378015507587338527226, 3.26041340519396988623860377630, 4.02675842500028177692737931271, 5.26792528833001663964848914339, 5.67434622530103733051830274612, 6.34753415638446266439330950306, 7.06256946951456400994554699672, 7.86762978860795334826875172203, 8.352908837777161780622767911779

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