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Properties

Label 2-490-245.103-c1-0-3
Degree $2$
Conductor $490$
Sign $-0.845 - 0.534i$
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 + (−2.93 + 0.554i)3-s + (−0.294 − 0.955i)4-s + (−1.81 − 1.30i)5-s + (1.29 − 2.68i)6-s + (2.61 + 0.393i)7-s + (0.943 + 0.330i)8-s + (5.49 − 2.15i)9-s + (2.12 − 0.681i)10-s + (−0.180 + 0.459i)11-s + (1.39 + 2.63i)12-s + (1.12 + 0.126i)13-s + (−1.86 + 1.87i)14-s + (6.04 + 2.83i)15-s + (−0.826 + 0.563i)16-s + (−6.29 − 0.235i)17-s + ⋯
L(s)  = 1  + (−0.419 + 0.568i)2-s + (−1.69 + 0.320i)3-s + (−0.147 − 0.477i)4-s + (−0.810 − 0.585i)5-s + (0.528 − 1.09i)6-s + (0.988 + 0.148i)7-s + (0.333 + 0.116i)8-s + (1.83 − 0.718i)9-s + (0.673 − 0.215i)10-s + (−0.0543 + 0.138i)11-s + (0.402 + 0.761i)12-s + (0.310 + 0.0350i)13-s + (−0.499 + 0.500i)14-s + (1.55 + 0.730i)15-s + (−0.206 + 0.140i)16-s + (−1.52 − 0.0571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.845 - 0.534i$
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: \((2,\ 490,\ (\ :1/2),\ -0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0830270 + 0.286863i\)
\(L(\frac12)\) \(\approx\) \(0.0830270 + 0.286863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.30i)T \)
7 \( 1 + (-2.61 - 0.393i)T \)
good3 \( 1 + (2.93 - 0.554i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.180 - 0.459i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-1.12 - 0.126i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (6.29 + 0.235i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (-3.57 + 6.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.316 - 8.44i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (2.67 - 0.609i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.23 - 3.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.12 - 3.76i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (-0.120 - 0.250i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (0.117 + 0.334i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-6.89 - 5.09i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (5.38 + 2.84i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.0710 - 0.948i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (1.87 - 6.09i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-11.1 - 2.99i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.10 - 4.84i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (10.6 - 7.85i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (-2.16 - 1.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.901 - 8.00i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (-2.25 - 5.75i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (4.37 + 4.37i)T + 97iT^{2} \)
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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

−11.27928204226310058670647361111, −10.80464007151902676885010758020, −9.426683235428647885344015856052, −8.714670825855997040691553548852, −7.44866748772875923161990915449, −6.85709022584740454352899252740, −5.44156010146324441926096868441, −5.05921821663536959798871361318, −4.10438438041206335467402645880, −1.27771924220304534903994498746, 0.29247739083956885680524481528, 1.90741007826149993496015309606, 3.87936669431583613050607200112, 4.78724662093923252718635039987, 5.96995601605961634129122317556, 6.98580660740373038995862232022, 7.74312747221488583341483849408, 8.742881141254995700525627977632, 10.28483776012707594522739064115, 10.92932298865946437219720847677

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