Properties

Label 2-490-5.4-c1-0-19
Degree $2$
Conductor $490$
Sign $-0.100 - 0.994i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
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 − 2.44i·3-s − 4-s + (−2.22 + 0.224i)5-s − 2.44·6-s + i·8-s − 2.99·9-s + (0.224 + 2.22i)10-s − 4.89·11-s + 2.44i·12-s + 4.44i·13-s + (0.550 + 5.44i)15-s + 16-s − 2i·17-s + 2.99i·18-s + 1.55·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.41i·3-s − 0.5·4-s + (−0.994 + 0.100i)5-s − 0.999·6-s + 0.353i·8-s − 0.999·9-s + (0.0710 + 0.703i)10-s − 1.47·11-s + 0.707i·12-s + 1.23i·13-s + (0.142 + 1.40i)15-s + 0.250·16-s − 0.485i·17-s + 0.707i·18-s + 0.355·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167912 + 0.185730i\)
\(L(\frac12)\) \(\approx\) \(0.167912 + 0.185730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (2.22 - 0.224i)T \)
7 \( 1 \)
good3 \( 1 + 2.44iT - 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 4.44iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 + 2.89iT - 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 + 8.89T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 1.10T + 41T^{2} \)
43 \( 1 - 0.898iT - 43T^{2} \)
47 \( 1 + 8.89iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 - 2.89iT - 73T^{2} \)
79 \( 1 + 6.89T + 79T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
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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

−10.66861485390635776776617658260, −9.344252357857677015491667574804, −8.349279124572727444829616895450, −7.51815889447922875993865195530, −6.98123750148877122484900315025, −5.55708497476079867512183240825, −4.32038634891721982666113220059, −2.96662214029900634387563728569, −1.84470633503726856444628060390, −0.14652777064952203964227770508, 3.18489074925452591269683327485, 3.98861477662556925968682259245, 5.13906010295543071750474090957, 5.60063092209570976289119478499, 7.40291975097973064529502854162, 7.931335787965834452649465330584, 8.899527850273962194991611529882, 9.830570585529713895969043715605, 10.67293811695615672360656784282, 11.21019923032026601338348259803

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