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Properties

Label 2-490-245.237-c1-0-4
Degree $2$
Conductor $490$
Sign $-0.502 - 0.864i$
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  + (−0.111 + 0.993i)2-s + (−0.990 − 0.346i)3-s + (−0.974 − 0.222i)4-s + (2.15 + 0.596i)5-s + (0.455 − 0.945i)6-s + (−2.49 − 0.870i)7-s + (0.330 − 0.943i)8-s + (−1.48 − 1.18i)9-s + (−0.834 + 2.07i)10-s + (2.76 + 3.46i)11-s + (0.888 + 0.558i)12-s + (−0.664 + 5.90i)13-s + (1.14 − 2.38i)14-s + (−1.92 − 1.33i)15-s + (0.900 + 0.433i)16-s + (3.94 + 2.47i)17-s + ⋯
L(s)  = 1  + (−0.0791 + 0.702i)2-s + (−0.572 − 0.200i)3-s + (−0.487 − 0.111i)4-s + (0.963 + 0.266i)5-s + (0.185 − 0.386i)6-s + (−0.944 − 0.329i)7-s + (0.116 − 0.333i)8-s + (−0.494 − 0.394i)9-s + (−0.263 + 0.656i)10-s + (0.832 + 1.04i)11-s + (0.256 + 0.161i)12-s + (−0.184 + 1.63i)13-s + (0.305 − 0.637i)14-s + (−0.497 − 0.345i)15-s + (0.225 + 0.108i)16-s + (0.955 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.466791 + 0.811142i\)
\(L(\frac12)\) \(\approx\) \(0.466791 + 0.811142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.111 - 0.993i)T \)
5 \( 1 + (-2.15 - 0.596i)T \)
7 \( 1 + (2.49 + 0.870i)T \)
good3 \( 1 + (0.990 + 0.346i)T + (2.34 + 1.87i)T^{2} \)
11 \( 1 + (-2.76 - 3.46i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.664 - 5.90i)T + (-12.6 - 2.89i)T^{2} \)
17 \( 1 + (-3.94 - 2.47i)T + (7.37 + 15.3i)T^{2} \)
19 \( 1 + 4.45T + 19T^{2} \)
23 \( 1 + (-2.28 - 3.63i)T + (-9.97 + 20.7i)T^{2} \)
29 \( 1 + (1.56 - 0.357i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + 6.34iT - 31T^{2} \)
37 \( 1 + (5.16 - 8.22i)T + (-16.0 - 33.3i)T^{2} \)
41 \( 1 + (-1.62 - 3.36i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (11.1 - 3.88i)T + (33.6 - 26.8i)T^{2} \)
47 \( 1 + (0.255 + 0.0288i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (1.53 + 2.44i)T + (-22.9 + 47.7i)T^{2} \)
59 \( 1 + (-7.57 - 3.64i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-7.17 + 1.63i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (6.53 + 6.53i)T + 67iT^{2} \)
71 \( 1 + (-2.14 + 9.39i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-14.3 + 1.62i)T + (71.1 - 16.2i)T^{2} \)
79 \( 1 + 8.09iT - 79T^{2} \)
83 \( 1 + (12.6 - 1.42i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-3.06 + 3.06i)T - 97iT^{2} \)
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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

−11.30298372000614536104572364945, −9.973190905286624050496350199676, −9.603097644921117789894571622086, −8.730382254961421486437168412538, −7.16373828704233057867350098819, −6.52678856764963256237986434456, −6.09746313619366315426947856597, −4.81335271419699973106747415588, −3.55200721700939327432985173775, −1.68554783045669435174795841573, 0.63500102415188991124108147423, 2.54744362602268805499674049608, 3.49677988744161553845467423438, 5.25541319768615056729863616346, 5.66226046079872714620275128612, 6.70962557289732289195259292406, 8.414061472680147096289929472011, 8.952072974036887688610288106687, 10.12634396895284492271645033583, 10.46402576740448501937637628965

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