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Properties

Label 2-425-85.29-c2-0-45
Degree $2$
Conductor $425$
Sign $0.168 + 0.985i$
Analytic cond. $11.5804$
Root an. cond. $3.40300$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.79 − 1.15i)2-s + (0.451 + 0.675i)3-s + (3.65 − 3.65i)4-s + (2.04 + 1.36i)6-s + (−1.53 − 7.70i)7-s + (1.34 − 3.25i)8-s + (3.19 − 7.70i)9-s + (4.48 − 6.71i)11-s + (4.11 + 0.818i)12-s + (−0.798 + 0.798i)13-s + (−13.2 − 19.7i)14-s + 9.98i·16-s + (15.7 − 6.50i)17-s − 25.2i·18-s + (−1.07 − 2.59i)19-s + ⋯
L(s)  = 1  + (1.39 − 0.579i)2-s + (0.150 + 0.225i)3-s + (0.913 − 0.913i)4-s + (0.340 + 0.227i)6-s + (−0.218 − 1.10i)7-s + (0.168 − 0.407i)8-s + (0.354 − 0.856i)9-s + (0.407 − 0.610i)11-s + (0.342 + 0.0682i)12-s + (−0.0614 + 0.0614i)13-s + (−0.943 − 1.41i)14-s + 0.624i·16-s + (0.923 − 0.382i)17-s − 1.40i·18-s + (−0.0566 − 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ 0.168 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.90754 - 2.45208i\)
\(L(\frac12)\) \(\approx\) \(2.90754 - 2.45208i\)
\(L(2)\) 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 \( 1 + (-15.7 + 6.50i)T \)
good2 \( 1 + (-2.79 + 1.15i)T + (2.82 - 2.82i)T^{2} \)
3 \( 1 + (-0.451 - 0.675i)T + (-3.44 + 8.31i)T^{2} \)
7 \( 1 + (1.53 + 7.70i)T + (-45.2 + 18.7i)T^{2} \)
11 \( 1 + (-4.48 + 6.71i)T + (-46.3 - 111. i)T^{2} \)
13 \( 1 + (0.798 - 0.798i)T - 169iT^{2} \)
19 \( 1 + (1.07 + 2.59i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (4.76 - 7.13i)T + (-202. - 488. i)T^{2} \)
29 \( 1 + (-0.599 + 3.01i)T + (-776. - 321. i)T^{2} \)
31 \( 1 + (7.13 + 10.6i)T + (-367. + 887. i)T^{2} \)
37 \( 1 + (-13.1 - 19.6i)T + (-523. + 1.26e3i)T^{2} \)
41 \( 1 + (-21.4 + 4.26i)T + (1.55e3 - 643. i)T^{2} \)
43 \( 1 + (-21.4 - 8.89i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (55.6 - 55.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (55.5 - 22.9i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-25.3 - 10.5i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-7.11 - 35.7i)T + (-3.43e3 + 1.42e3i)T^{2} \)
67 \( 1 + 117.T + 4.48e3T^{2} \)
71 \( 1 + (-88.1 + 58.8i)T + (1.92e3 - 4.65e3i)T^{2} \)
73 \( 1 + (11.9 - 59.8i)T + (-4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (52.9 - 79.2i)T + (-2.38e3 - 5.76e3i)T^{2} \)
83 \( 1 + (-45.4 - 109. i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-61.4 + 61.4i)T - 7.92e3iT^{2} \)
97 \( 1 + (-24.1 - 4.79i)T + (8.69e3 + 3.60e3i)T^{2} \)
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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

−11.02625045075949185107998482535, −10.08517361430368517628941479962, −9.232335978239345795528336089080, −7.84258199053840593679039292929, −6.68462201801946979429367425944, −5.82150467954655644783152871568, −4.54081107688023062309384415899, −3.78518196273588566468569802655, −3.00929409926633682035091963428, −1.12953359683853424700937573714, 2.04882447660668920248789281560, 3.29391872882597846464075055904, 4.52762545177539524156238516110, 5.40104326260852999497222348580, 6.22691518264866425521185480291, 7.21984154805909362170191107571, 8.107084781380685809103032660743, 9.308383702392335795203669252490, 10.30337519072356289247203513616, 11.62107359287636102983210529100

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