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Properties

Label 2-490-245.103-c1-0-26
Degree $2$
Conductor $490$
Sign $-0.410 + 0.911i$
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 + (2.22 − 0.420i)3-s + (−0.294 − 0.955i)4-s + (−1.80 − 1.32i)5-s + (0.981 − 2.03i)6-s + (−1.72 − 2.00i)7-s + (−0.943 − 0.330i)8-s + (1.96 − 0.771i)9-s + (−2.13 + 0.667i)10-s + (0.366 − 0.934i)11-s + (−1.05 − 1.99i)12-s + (6.34 + 0.714i)13-s + (−2.63 + 0.195i)14-s + (−4.56 − 2.17i)15-s + (−0.826 + 0.563i)16-s + (−3.80 − 0.142i)17-s + ⋯
L(s)  = 1  + (0.419 − 0.568i)2-s + (1.28 − 0.242i)3-s + (−0.147 − 0.477i)4-s + (−0.806 − 0.590i)5-s + (0.400 − 0.831i)6-s + (−0.651 − 0.758i)7-s + (−0.333 − 0.116i)8-s + (0.655 − 0.257i)9-s + (−0.674 + 0.211i)10-s + (0.110 − 0.281i)11-s + (−0.304 − 0.577i)12-s + (1.75 + 0.198i)13-s + (−0.705 + 0.0523i)14-s + (−1.17 − 0.561i)15-s + (−0.206 + 0.140i)16-s + (−0.923 − 0.0345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.410 + 0.911i$
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: \((2,\ 490,\ (\ :1/2),\ -0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15611 - 1.78836i\)
\(L(\frac12)\) \(\approx\) \(1.15611 - 1.78836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.32i)T \)
7 \( 1 + (1.72 + 2.00i)T \)
good3 \( 1 + (-2.22 + 0.420i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (-0.366 + 0.934i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-6.34 - 0.714i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (3.80 + 0.142i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-0.765 + 1.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0770 + 2.05i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (-1.02 + 0.234i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.10 + 0.639i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.81 + 3.60i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-1.69 - 3.51i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-1.56 - 4.46i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-5.71 - 4.21i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (-7.21 - 3.81i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.419 + 5.59i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.70 + 8.76i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-7.35 - 1.97i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.14 - 13.7i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (7.78 - 5.74i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (5.99 + 3.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.849 - 7.54i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (3.92 + 10.0i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (4.87 + 4.87i)T + 97iT^{2} \)
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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

−10.94473147407265774341654603166, −9.614810233547757572783551293428, −8.828009602013960927754492396368, −8.235051093521800553862642396144, −7.14227491169267917828530430381, −6.04173972769244877855998266784, −4.32061495884171232644006315518, −3.76863813581180725369806670939, −2.75595036159829668546509708944, −1.06860261380455261779379067113, 2.53908472294915723001778900123, 3.49246694286423706138062930534, 4.13131347328659306564270990561, 5.80956875266189149059651356120, 6.69054710048468645912605454720, 7.72261406318271717714843784247, 8.609078227999465316002116973703, 9.003543765398301914359117914144, 10.23578116868055352478906834720, 11.34549945899577619144446413481

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