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Properties

Label 2-490-5.4-c3-0-33
Degree $2$
Conductor $490$
Sign $0.990 - 0.137i$
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 − 4.69i·3-s − 4·4-s + (1.54 + 11.0i)5-s + 9.38·6-s − 8i·8-s + 4.97·9-s + (−22.1 + 3.08i)10-s + 1.37·11-s + 18.7i·12-s − 69.4i·13-s + (51.9 − 7.23i)15-s + 16·16-s + 26.3i·17-s + 9.94i·18-s − 14.1·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.903i·3-s − 0.5·4-s + (0.137 + 0.990i)5-s + 0.638·6-s − 0.353i·8-s + 0.184·9-s + (−0.700 + 0.0975i)10-s + 0.0377·11-s + 0.451i·12-s − 1.48i·13-s + (0.894 − 0.124i)15-s + 0.250·16-s + 0.375i·17-s + 0.130i·18-s − 0.171·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.990 - 0.137i$
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: \((2,\ 490,\ (\ :3/2),\ 0.990 - 0.137i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.922974402\)
\(L(\frac12)\) \(\approx\) \(1.922974402\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.54 - 11.0i)T \)
7 \( 1 \)
good3 \( 1 + 4.69iT - 27T^{2} \)
11 \( 1 - 1.37T + 1.33e3T^{2} \)
13 \( 1 + 69.4iT - 2.19e3T^{2} \)
17 \( 1 - 26.3iT - 4.91e3T^{2} \)
19 \( 1 + 14.1T + 6.85e3T^{2} \)
23 \( 1 - 29.6iT - 1.21e4T^{2} \)
29 \( 1 - 75.5T + 2.43e4T^{2} \)
31 \( 1 - 245.T + 2.97e4T^{2} \)
37 \( 1 - 245. iT - 5.06e4T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 - 194. iT - 7.95e4T^{2} \)
47 \( 1 + 375. iT - 1.03e5T^{2} \)
53 \( 1 + 511. iT - 1.48e5T^{2} \)
59 \( 1 - 656.T + 2.05e5T^{2} \)
61 \( 1 - 809.T + 2.26e5T^{2} \)
67 \( 1 + 220. iT - 3.00e5T^{2} \)
71 \( 1 - 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + 898. iT - 3.89e5T^{2} \)
79 \( 1 - 708.T + 4.93e5T^{2} \)
83 \( 1 - 419. iT - 5.71e5T^{2} \)
89 \( 1 - 393.T + 7.04e5T^{2} \)
97 \( 1 - 152. iT - 9.12e5T^{2} \)
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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

−10.35547856178664730462694508029, −9.838089721904215266046490905580, −8.263616397356260856311112236962, −7.82216525601184572769646403449, −6.77934023449148098905336434047, −6.29422098615312790325053664968, −5.16135361450562971914713905435, −3.65639807530110086812387385272, −2.41784410181953632335986686696, −0.811291039078910323122177489371, 0.987732365968233515949791167575, 2.33778822232525759418347264204, 4.01557236292421006878960400206, 4.44855155739958860963780626898, 5.43394901517317596749167768328, 6.80393507330204235967262907837, 8.204917400092718134762361174896, 9.160163096596568755901246601789, 9.548097415539605075513241377181, 10.44963730567606549912361436780

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