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Properties

Label 2-490-245.237-c1-0-9
Degree $2$
Conductor $490$
Sign $0.682 + 0.730i$
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 + (−2.73 − 0.958i)3-s + (−0.974 − 0.222i)4-s + (2.16 − 0.572i)5-s + (−1.25 + 2.61i)6-s + (2.33 + 1.23i)7-s + (−0.330 + 0.943i)8-s + (4.24 + 3.38i)9-s + (−0.327 − 2.21i)10-s + (0.646 + 0.811i)11-s + (2.45 + 1.54i)12-s + (−0.790 + 7.01i)13-s + (1.49 − 2.18i)14-s + (−6.47 − 0.502i)15-s + (0.900 + 0.433i)16-s + (1.68 + 1.05i)17-s + ⋯
L(s)  = 1  + (0.0791 − 0.702i)2-s + (−1.58 − 0.553i)3-s + (−0.487 − 0.111i)4-s + (0.966 − 0.256i)5-s + (−0.514 + 1.06i)6-s + (0.883 + 0.468i)7-s + (−0.116 + 0.333i)8-s + (1.41 + 1.12i)9-s + (−0.103 − 0.699i)10-s + (0.195 + 0.244i)11-s + (0.709 + 0.445i)12-s + (−0.219 + 1.94i)13-s + (0.399 − 0.583i)14-s + (−1.67 − 0.129i)15-s + (0.225 + 0.108i)16-s + (0.407 + 0.256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.682 + 0.730i$
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: \((2,\ 490,\ (\ :1/2),\ 0.682 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01500 - 0.440981i\)
\(L(\frac12)\) \(\approx\) \(1.01500 - 0.440981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 0.572i)T \)
7 \( 1 + (-2.33 - 1.23i)T \)
good3 \( 1 + (2.73 + 0.958i)T + (2.34 + 1.87i)T^{2} \)
11 \( 1 + (-0.646 - 0.811i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.790 - 7.01i)T + (-12.6 - 2.89i)T^{2} \)
17 \( 1 + (-1.68 - 1.05i)T + (7.37 + 15.3i)T^{2} \)
19 \( 1 - 5.66T + 19T^{2} \)
23 \( 1 + (4.02 + 6.41i)T + (-9.97 + 20.7i)T^{2} \)
29 \( 1 + (-3.66 + 0.835i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + 0.443iT - 31T^{2} \)
37 \( 1 + (-2.01 + 3.21i)T + (-16.0 - 33.3i)T^{2} \)
41 \( 1 + (-3.05 - 6.33i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (4.73 - 1.65i)T + (33.6 - 26.8i)T^{2} \)
47 \( 1 + (12.9 + 1.45i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (0.823 + 1.31i)T + (-22.9 + 47.7i)T^{2} \)
59 \( 1 + (-3.58 - 1.72i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-2.28 + 0.522i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \)
71 \( 1 + (-2.52 + 11.0i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.463 + 0.0522i)T + (71.1 - 16.2i)T^{2} \)
79 \( 1 + 5.53iT - 79T^{2} \)
83 \( 1 + (-5.88 + 0.663i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (7.43 - 9.32i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-2.10 + 2.10i)T - 97iT^{2} \)
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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

−11.16255519122456761828007174818, −10.09765277282856640837170010459, −9.411543916832981741097894974501, −8.212136193177787121343265459573, −6.81319159891055400455684286717, −6.08767589615550720780185632735, −5.09535664420430717934746123701, −4.48911626723814019911291986515, −2.15030804750136326724433885198, −1.28996411101788836345574786178, 1.00579096185506717242409085428, 3.47019623579184901882260566232, 5.09760340828015355343655700204, 5.31149307210196695206656371282, 6.15084854513663029295395587822, 7.24706778164283316332628619804, 8.146053294377780344922574190956, 9.767340597479478035921173795460, 10.07678608744853775242405669991, 11.01583601012785408687426043643

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