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Properties

Label 2-490-5.4-c3-0-30
Degree $2$
Conductor $490$
Sign $0.954 - 0.298i$
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 + 4.49i·3-s − 4·4-s + (3.34 + 10.6i)5-s + 8.98·6-s + 8i·8-s + 6.79·9-s + (21.3 − 6.68i)10-s + 13.9·11-s − 17.9i·12-s − 78.0i·13-s + (−47.9 + 15.0i)15-s + 16·16-s − 105. i·17-s − 13.5i·18-s + 152.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.865i·3-s − 0.5·4-s + (0.298 + 0.954i)5-s + 0.611·6-s + 0.353i·8-s + 0.251·9-s + (0.674 − 0.211i)10-s + 0.381·11-s − 0.432i·12-s − 1.66i·13-s + (−0.825 + 0.258i)15-s + 0.250·16-s − 1.49i·17-s − 0.177i·18-s + 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.954 - 0.298i$
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: \((2,\ 490,\ (\ :3/2),\ 0.954 - 0.298i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.188677939\)
\(L(\frac12)\) \(\approx\) \(2.188677939\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.34 - 10.6i)T \)
7 \( 1 \)
good3 \( 1 - 4.49iT - 27T^{2} \)
11 \( 1 - 13.9T + 1.33e3T^{2} \)
13 \( 1 + 78.0iT - 2.19e3T^{2} \)
17 \( 1 + 105. iT - 4.91e3T^{2} \)
19 \( 1 - 152.T + 6.85e3T^{2} \)
23 \( 1 - 108. iT - 1.21e4T^{2} \)
29 \( 1 - 109.T + 2.43e4T^{2} \)
31 \( 1 - 197.T + 2.97e4T^{2} \)
37 \( 1 - 254. iT - 5.06e4T^{2} \)
41 \( 1 + 193.T + 6.89e4T^{2} \)
43 \( 1 - 41.5iT - 7.95e4T^{2} \)
47 \( 1 - 109. iT - 1.03e5T^{2} \)
53 \( 1 + 11.0iT - 1.48e5T^{2} \)
59 \( 1 - 192.T + 2.05e5T^{2} \)
61 \( 1 - 34.9T + 2.26e5T^{2} \)
67 \( 1 - 374. iT - 3.00e5T^{2} \)
71 \( 1 + 575.T + 3.57e5T^{2} \)
73 \( 1 - 522. iT - 3.89e5T^{2} \)
79 \( 1 - 456.T + 4.93e5T^{2} \)
83 \( 1 + 773. iT - 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 - 1.03e3iT - 9.12e5T^{2} \)
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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

−10.33944886206697475247816532843, −9.954754375056372833296968704730, −9.316053647707227697827312549965, −7.903128790915453259299786469219, −7.01741238896373229562325381074, −5.59148301165876049805982879166, −4.80996206698386832491317830059, −3.35723451973919390736651592830, −2.92565639051176048539466516859, −1.05141815836192003467139145978, 0.936862910912777676440939060713, 1.89995756871108186272447206200, 3.98064713773730332455769356847, 4.86971422838211409961443816981, 6.12522353960834640643375352258, 6.73630020969627490136247588745, 7.74283205597533552376643744072, 8.594564771535650827031550175727, 9.350634458326887229377306009325, 10.25639480650331332948764550400

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