Properties

Label 2-490-5.4-c1-0-5
Degree $2$
Conductor $490$
Sign $0.948 - 0.316i$
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 − 4-s + (0.707 + 2.12i)5-s + i·8-s + 3·9-s + (2.12 − 0.707i)10-s − 4·11-s + 4.24i·13-s + 16-s + 4.24i·17-s − 3i·18-s + 5.65·19-s + (−0.707 − 2.12i)20-s + 4i·22-s + (−3.99 + 3i)25-s + 4.24·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.316 + 0.948i)5-s + 0.353i·8-s + 9-s + (0.670 − 0.223i)10-s − 1.20·11-s + 1.17i·13-s + 0.250·16-s + 1.02i·17-s − 0.707i·18-s + 1.29·19-s + (−0.158 − 0.474i)20-s + 0.852i·22-s + (−0.799 + 0.600i)25-s + 0.832·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.948 - 0.316i$
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: \((2,\ 490,\ (\ :1/2),\ 0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37302 + 0.222810i\)
\(L(\frac12)\) \(\approx\) \(1.37302 + 0.222810i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 2.12i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 + 4.24iT - 97T^{2} \)
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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

−10.87769819414411735077786691014, −10.14508993560069564312402724311, −9.669660073948393136817974067973, −8.357884000437047869321260788229, −7.34587468893661900469756302929, −6.46173115894001115660171445820, −5.19417461552861679916861536910, −4.04848521280924102809669139197, −2.88601456490915320973124656599, −1.70737242332770347547175238758, 0.928347578982045791759048188999, 2.92698308589819471079337425120, 4.61712705212728812186918085705, 5.17301000859698927140033500937, 6.16267390552995034854766583844, 7.62511545832618289282051844766, 7.85174869683744922961984569459, 9.173959814220997578014290342030, 9.835684290768663595707838971062, 10.64799302755001172916297119123

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