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Properties

Label 2-490-5.4-c3-0-38
Degree $2$
Conductor $490$
Sign $0.601 - 0.799i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 0.537i·3-s − 4·4-s + (8.93 + 6.72i)5-s + 1.07·6-s − 8i·8-s + 26.7·9-s + (−13.4 + 17.8i)10-s + 14.1·11-s + 2.15i·12-s − 13.9i·13-s + (3.61 − 4.80i)15-s + 16·16-s − 65.7i·17-s + 53.4i·18-s + 95.1·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.103i·3-s − 0.5·4-s + (0.799 + 0.601i)5-s + 0.0731·6-s − 0.353i·8-s + 0.989·9-s + (−0.425 + 0.565i)10-s + 0.388·11-s + 0.0517i·12-s − 0.296i·13-s + (0.0622 − 0.0827i)15-s + 0.250·16-s − 0.938i·17-s + 0.699i·18-s + 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.601 - 0.799i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ 0.601 - 0.799i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.470212457\)
\(L(\frac12)\) \(\approx\) \(2.470212457\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-8.93 - 6.72i)T \)
7 \( 1 \)
good3 \( 1 + 0.537iT - 27T^{2} \)
11 \( 1 - 14.1T + 1.33e3T^{2} \)
13 \( 1 + 13.9iT - 2.19e3T^{2} \)
17 \( 1 + 65.7iT - 4.91e3T^{2} \)
19 \( 1 - 95.1T + 6.85e3T^{2} \)
23 \( 1 + 69.4iT - 1.21e4T^{2} \)
29 \( 1 + 127.T + 2.43e4T^{2} \)
31 \( 1 - 98.3T + 2.97e4T^{2} \)
37 \( 1 + 287. iT - 5.06e4T^{2} \)
41 \( 1 - 310.T + 6.89e4T^{2} \)
43 \( 1 + 197. iT - 7.95e4T^{2} \)
47 \( 1 - 538. iT - 1.03e5T^{2} \)
53 \( 1 - 314. iT - 1.48e5T^{2} \)
59 \( 1 - 242.T + 2.05e5T^{2} \)
61 \( 1 - 440.T + 2.26e5T^{2} \)
67 \( 1 - 858. iT - 3.00e5T^{2} \)
71 \( 1 - 142.T + 3.57e5T^{2} \)
73 \( 1 - 459. iT - 3.89e5T^{2} \)
79 \( 1 + 1.19e3T + 4.93e5T^{2} \)
83 \( 1 + 403. iT - 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 166. iT - 9.12e5T^{2} \)
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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

−10.45528132451164975274792183836, −9.660433023261939876067057638820, −9.052030165446623238840371229724, −7.58249700610856992009730393221, −7.10975451742883841211730640584, −6.11197423544459011069542010979, −5.22589283135170458219657829305, −4.01153503677147409034673174998, −2.61287922746695305495798346054, −1.05424298120639048162498049394, 1.08985987295734178183529712152, 1.95165236190713779567781901435, 3.53467720715571146039609419316, 4.55858456945987720955990012030, 5.52165028048710734055104977363, 6.63878815033264641738315844990, 7.87001879995715321354103527432, 8.940791405329662645395071668850, 9.721238956956128778734925948938, 10.18138948160193172379171571670

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