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Properties

Label 2-425-5.4-c3-0-69
Degree $2$
Conductor $425$
Sign $0.447 - 0.894i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
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  − 3.16i·2-s − 9.73i·3-s − 2.01·4-s − 30.8·6-s + 10.5i·7-s − 18.9i·8-s − 67.8·9-s − 15.1·11-s + 19.6i·12-s − 89.5i·13-s + 33.3·14-s − 76.0·16-s − 17i·17-s + 214. i·18-s − 1.82·19-s + ⋯
L(s)  = 1  − 1.11i·2-s − 1.87i·3-s − 0.252·4-s − 2.09·6-s + 0.568i·7-s − 0.836i·8-s − 2.51·9-s − 0.414·11-s + 0.473i·12-s − 1.91i·13-s + 0.636·14-s − 1.18·16-s − 0.242i·17-s + 2.81i·18-s − 0.0220·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.167784015\)
\(L(\frac12)\) \(\approx\) \(1.167784015\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + 17iT \)
good2 \( 1 + 3.16iT - 8T^{2} \)
3 \( 1 + 9.73iT - 27T^{2} \)
7 \( 1 - 10.5iT - 343T^{2} \)
11 \( 1 + 15.1T + 1.33e3T^{2} \)
13 \( 1 + 89.5iT - 2.19e3T^{2} \)
19 \( 1 + 1.82T + 6.85e3T^{2} \)
23 \( 1 - 170. iT - 1.21e4T^{2} \)
29 \( 1 + 53.6T + 2.43e4T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 - 94.8iT - 5.06e4T^{2} \)
41 \( 1 - 374.T + 6.89e4T^{2} \)
43 \( 1 - 101. iT - 7.95e4T^{2} \)
47 \( 1 + 489. iT - 1.03e5T^{2} \)
53 \( 1 + 84.6iT - 1.48e5T^{2} \)
59 \( 1 + 467.T + 2.05e5T^{2} \)
61 \( 1 + 836.T + 2.26e5T^{2} \)
67 \( 1 - 47.4iT - 3.00e5T^{2} \)
71 \( 1 - 517.T + 3.57e5T^{2} \)
73 \( 1 + 398. iT - 3.89e5T^{2} \)
79 \( 1 + 856.T + 4.93e5T^{2} \)
83 \( 1 + 946. iT - 5.71e5T^{2} \)
89 \( 1 - 765.T + 7.04e5T^{2} \)
97 \( 1 - 124. iT - 9.12e5T^{2} \)
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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

−10.30829690996432119780274277680, −9.105278711589299979896163619252, −7.949615696427502313457641176171, −7.42127650436598864269329544650, −6.21254148234651631107421725737, −5.41333780727177049165576659323, −3.20419416598168237899651621749, −2.55114690348990730881620473249, −1.42010519696998312589892517685, −0.37352713833955578900936499307, 2.58919305078461590428585189613, 4.21337788274443501308540401523, 4.60464153821501434997297191028, 5.82350846886286880371075683567, 6.67492405669201261923197665222, 7.905452241322425063279167010077, 8.883199750278427651722006974493, 9.485861887060824170861856839642, 10.67048327701948563483164154861, 11.04784138620443582609946427812

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