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Properties

Label 2-425-17.10-c2-0-23
Degree $2$
Conductor $425$
Sign $0.792 - 0.609i$
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  + (0.552 − 1.33i)2-s + (3.95 + 2.64i)3-s + (1.35 + 1.35i)4-s + (5.71 − 3.81i)6-s + (−8.88 − 1.76i)7-s + (7.89 − 3.26i)8-s + (5.22 + 12.6i)9-s + (11.2 + 16.7i)11-s + (1.77 + 8.93i)12-s + (−4.55 + 4.55i)13-s + (−7.26 + 10.8i)14-s − 4.67i·16-s + (−10.6 + 13.2i)17-s + 19.6·18-s + (4.69 − 11.3i)19-s + ⋯
L(s)  = 1  + (0.276 − 0.666i)2-s + (1.31 + 0.881i)3-s + (0.338 + 0.338i)4-s + (0.952 − 0.636i)6-s + (−1.26 − 0.252i)7-s + (0.986 − 0.408i)8-s + (0.580 + 1.40i)9-s + (1.01 + 1.52i)11-s + (0.148 + 0.744i)12-s + (−0.350 + 0.350i)13-s + (−0.518 + 0.776i)14-s − 0.291i·16-s + (−0.624 + 0.780i)17-s + 1.09·18-s + (0.247 − 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.792 - 0.609i$
Analytic conductor: \(11.5804\)
Root analytic conductor: \(3.40300\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1),\ 0.792 - 0.609i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.07387 + 1.04550i\)
\(L(\frac12)\) \(\approx\) \(3.07387 + 1.04550i\)
\(L(2)\) 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 \( 1 + (10.6 - 13.2i)T \)
good2 \( 1 + (-0.552 + 1.33i)T + (-2.82 - 2.82i)T^{2} \)
3 \( 1 + (-3.95 - 2.64i)T + (3.44 + 8.31i)T^{2} \)
7 \( 1 + (8.88 + 1.76i)T + (45.2 + 18.7i)T^{2} \)
11 \( 1 + (-11.2 - 16.7i)T + (-46.3 + 111. i)T^{2} \)
13 \( 1 + (4.55 - 4.55i)T - 169iT^{2} \)
19 \( 1 + (-4.69 + 11.3i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-27.7 + 18.5i)T + (202. - 488. i)T^{2} \)
29 \( 1 + (1.76 + 8.88i)T + (-776. + 321. i)T^{2} \)
31 \( 1 + (-5.24 + 7.84i)T + (-367. - 887. i)T^{2} \)
37 \( 1 + (30.8 + 20.5i)T + (523. + 1.26e3i)T^{2} \)
41 \( 1 + (20.0 + 3.98i)T + (1.55e3 + 643. i)T^{2} \)
43 \( 1 + (-1.42 - 3.44i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + (-59.9 + 59.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (31.5 - 76.2i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (10.2 - 4.23i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-13.7 + 69.1i)T + (-3.43e3 - 1.42e3i)T^{2} \)
67 \( 1 - 53.5iT - 4.48e3T^{2} \)
71 \( 1 + (-32.2 - 21.5i)T + (1.92e3 + 4.65e3i)T^{2} \)
73 \( 1 + (-41.3 + 8.22i)T + (4.92e3 - 2.03e3i)T^{2} \)
79 \( 1 + (29.3 + 43.9i)T + (-2.38e3 + 5.76e3i)T^{2} \)
83 \( 1 + (-2.25 - 0.933i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (109. + 109. i)T + 7.92e3iT^{2} \)
97 \( 1 + (14.7 + 74.2i)T + (-8.69e3 + 3.60e3i)T^{2} \)
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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

−10.85527567213730549771678230527, −10.03983405422455816675634841358, −9.428503733039617992155603813979, −8.668517640741336879808259539571, −7.24500083231276778845571747002, −6.72898801560895887364570267258, −4.52040715095146512687498069682, −3.95209379301754583860716266652, −2.98947232547132124490550368764, −2.03630585800497762105665362280, 1.19027730307004702345372518739, 2.74804979882653666252120676056, 3.52412994920164465715922800145, 5.41995392035974614403368240552, 6.51094371667447195731835586508, 6.94896236966689245860754824726, 7.988944780644367763031301120529, 8.942100008729108593739634318450, 9.574806155114557023223170311714, 10.90277655659647243666735340631

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