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Properties

Label 2-425-5.4-c3-0-1
Degree $2$
Conductor $425$
Sign $0.894 - 0.447i$
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  + 5.03i·2-s + 8.47i·3-s − 17.3·4-s − 42.6·6-s − 3.81i·7-s − 46.9i·8-s − 44.8·9-s − 52.3·11-s − 146. i·12-s − 8.06i·13-s + 19.2·14-s + 97.5·16-s + 17i·17-s − 225. i·18-s + 66.5·19-s + ⋯
L(s)  = 1  + 1.77i·2-s + 1.63i·3-s − 2.16·4-s − 2.90·6-s − 0.206i·7-s − 2.07i·8-s − 1.66·9-s − 1.43·11-s − 3.53i·12-s − 0.171i·13-s + 0.366·14-s + 1.52·16-s + 0.242i·17-s − 2.95i·18-s + 0.803·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.894 - 0.447i$
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: \((2,\ 425,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2248739685\)
\(L(\frac12)\) \(\approx\) \(0.2248739685\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.03iT - 8T^{2} \)
3 \( 1 - 8.47iT - 27T^{2} \)
7 \( 1 + 3.81iT - 343T^{2} \)
11 \( 1 + 52.3T + 1.33e3T^{2} \)
13 \( 1 + 8.06iT - 2.19e3T^{2} \)
19 \( 1 - 66.5T + 6.85e3T^{2} \)
23 \( 1 - 180. iT - 1.21e4T^{2} \)
29 \( 1 - 41.2T + 2.43e4T^{2} \)
31 \( 1 + 34.9T + 2.97e4T^{2} \)
37 \( 1 + 130. iT - 5.06e4T^{2} \)
41 \( 1 + 17.9T + 6.89e4T^{2} \)
43 \( 1 - 277. iT - 7.95e4T^{2} \)
47 \( 1 + 463. iT - 1.03e5T^{2} \)
53 \( 1 + 329. iT - 1.48e5T^{2} \)
59 \( 1 + 678.T + 2.05e5T^{2} \)
61 \( 1 - 340.T + 2.26e5T^{2} \)
67 \( 1 + 15.3iT - 3.00e5T^{2} \)
71 \( 1 + 670.T + 3.57e5T^{2} \)
73 \( 1 - 193. iT - 3.89e5T^{2} \)
79 \( 1 + 1.08e3T + 4.93e5T^{2} \)
83 \( 1 + 865. iT - 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 - 379. iT - 9.12e5T^{2} \)
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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

−11.45461047210738229528659475286, −10.38413192332210870199411076316, −9.746457741438977894566104181589, −8.914492830799772135142786925585, −7.994196765295778289398744891610, −7.22505765040479956245108264050, −5.69798356462873613843304592610, −5.30629802115570922014546487918, −4.36482635754293517491777256422, −3.27268641822377700076553786486, 0.082260078972871141180231633733, 1.16481525190972208209689148391, 2.36654593432882710030702277971, 2.93183317123516212365976254532, 4.64355469687558047105653199221, 5.81170233445064569799351595695, 7.16521568764293099932164693137, 8.120663989289146465774140741793, 8.919361222159977676564276232174, 10.11788709745410239136218288886

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