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Properties

Label 2-490-5.4-c3-0-18
Degree $2$
Conductor $490$
Sign $-0.959 - 0.281i$
Analytic cond. $28.9109$
Root an. cond. $5.37688$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 6.00i·3-s − 4·4-s + (3.14 − 10.7i)5-s − 12.0·6-s − 8i·8-s − 9.02·9-s + (21.4 + 6.28i)10-s + 9.14·11-s − 24.0i·12-s − 32.7i·13-s + (64.4 + 18.8i)15-s + 16·16-s + 109. i·17-s − 18.0i·18-s + 22.2·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.15i·3-s − 0.5·4-s + (0.281 − 0.959i)5-s − 0.816·6-s − 0.353i·8-s − 0.334·9-s + (0.678 + 0.198i)10-s + 0.250·11-s − 0.577i·12-s − 0.697i·13-s + (1.10 + 0.324i)15-s + 0.250·16-s + 1.56i·17-s − 0.236i·18-s + 0.268·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.959 - 0.281i$
Analytic conductor: \(28.9109\)
Root analytic conductor: \(5.37688\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :3/2),\ -0.959 - 0.281i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.550901188\)
\(L(\frac12)\) \(\approx\) \(1.550901188\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-3.14 + 10.7i)T \)
7 \( 1 \)
good3 \( 1 - 6.00iT - 27T^{2} \)
11 \( 1 - 9.14T + 1.33e3T^{2} \)
13 \( 1 + 32.7iT - 2.19e3T^{2} \)
17 \( 1 - 109. iT - 4.91e3T^{2} \)
19 \( 1 - 22.2T + 6.85e3T^{2} \)
23 \( 1 - 191. iT - 1.21e4T^{2} \)
29 \( 1 + 190.T + 2.43e4T^{2} \)
31 \( 1 - 35.6T + 2.97e4T^{2} \)
37 \( 1 - 57.7iT - 5.06e4T^{2} \)
41 \( 1 - 39.7T + 6.89e4T^{2} \)
43 \( 1 - 323. iT - 7.95e4T^{2} \)
47 \( 1 - 57.1iT - 1.03e5T^{2} \)
53 \( 1 - 529. iT - 1.48e5T^{2} \)
59 \( 1 - 766.T + 2.05e5T^{2} \)
61 \( 1 - 524.T + 2.26e5T^{2} \)
67 \( 1 - 370. iT - 3.00e5T^{2} \)
71 \( 1 + 722.T + 3.57e5T^{2} \)
73 \( 1 + 829. iT - 3.89e5T^{2} \)
79 \( 1 + 494.T + 4.93e5T^{2} \)
83 \( 1 + 1.24e3iT - 5.71e5T^{2} \)
89 \( 1 + 20.7T + 7.04e5T^{2} \)
97 \( 1 - 1.01e3iT - 9.12e5T^{2} \)
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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

−10.66380098989827825326361038917, −9.817308370570094887679484750705, −9.245116996022137174145925089308, −8.370367396491930308387833959633, −7.47549790802065044953914785527, −5.95135439707949144964449153181, −5.37604294913965765923650694480, −4.34239955215916771519551951514, −3.55911615204368888538593530359, −1.38585256997986744666990269479, 0.50980479715410204338671273630, 1.94372214355271280225277078300, 2.70584680432516026505822971871, 4.07171086802850006858970623665, 5.49999531724846362419492030125, 6.80255543759415992287796013577, 7.09695993702406719048613150348, 8.329884878139361518294820094682, 9.423550491303572333059099113832, 10.19709614553363586273470667941

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