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Properties

Label 2-490-245.103-c1-0-2
Degree $2$
Conductor $490$
Sign $-0.444 - 0.895i$
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  + (0.593 − 0.804i)2-s + (0.000603 − 0.000114i)3-s + (−0.294 − 0.955i)4-s + (−1.17 + 1.90i)5-s + (0.000266 − 0.000553i)6-s + (−2.50 + 0.846i)7-s + (−0.943 − 0.330i)8-s + (−2.79 + 1.09i)9-s + (0.837 + 2.07i)10-s + (0.156 − 0.398i)11-s + (−0.000286 − 0.000542i)12-s + (−0.598 − 0.0674i)13-s + (−0.806 + 2.51i)14-s + (−0.000488 + 0.00128i)15-s + (−0.826 + 0.563i)16-s + (−1.75 − 0.0657i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (0.000348 − 6.59e−5i)3-s + (−0.147 − 0.477i)4-s + (−0.523 + 0.851i)5-s + (0.000108 − 0.000225i)6-s + (−0.947 + 0.320i)7-s + (−0.333 − 0.116i)8-s + (−0.930 + 0.365i)9-s + (0.264 + 0.655i)10-s + (0.0471 − 0.120i)11-s + (−8.28e−5 − 0.000156i)12-s + (−0.166 − 0.0187i)13-s + (−0.215 + 0.673i)14-s + (−0.000126 + 0.000331i)15-s + (−0.206 + 0.140i)16-s + (−0.426 − 0.0159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264172 + 0.426018i\)
\(L(\frac12)\) \(\approx\) \(0.264172 + 0.426018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.593 + 0.804i)T \)
5 \( 1 + (1.17 - 1.90i)T \)
7 \( 1 + (2.50 - 0.846i)T \)
good3 \( 1 + (-0.000603 + 0.000114i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-0.156 + 0.398i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (0.598 + 0.0674i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (1.75 + 0.0657i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (3.69 - 6.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0107 - 0.287i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (1.89 - 0.433i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.61 + 0.931i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.88 + 3.11i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.46 - 3.04i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.173 - 0.496i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-4.38 - 3.23i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (9.71 + 5.13i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.483 - 6.45i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (1.55 - 5.03i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (10.1 + 2.70i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.33 - 10.2i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-7.92 + 5.85i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (5.95 + 3.43i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.147 - 1.30i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-1.25 - 3.18i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-0.360 - 0.360i)T + 97iT^{2} \)
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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

−11.24180688542353315686864114278, −10.55414530496907717773011328680, −9.698067501677754575029317517944, −8.633181226494545044197682376132, −7.63185671790750714760654504963, −6.35605443854256128865391636606, −5.79857248057289061895664248576, −4.25510805583197632686730367388, −3.24929411487520545811466524180, −2.37043676146434591746123775548, 0.24551706707519058719254153369, 2.81680491216120741960536158559, 4.00447167624955263737876429326, 4.90926196083468220262782326193, 6.07204509994753808949969505379, 6.86679423849273671870256245273, 7.928341286072608309607660110808, 8.902795793213728700048702389977, 9.396551581049829480312109258643, 10.84238999288703440749755084425

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