Properties

Label 2-425-17.16-c3-0-43
Degree $2$
Conductor $425$
Sign $0.973 + 0.230i$
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  − 2.37·2-s + 4.44i·3-s − 2.37·4-s − 10.5i·6-s + 14.9i·7-s + 24.6·8-s + 7.23·9-s − 31.1i·11-s − 10.5i·12-s + 5.21·13-s − 35.5i·14-s − 39.3·16-s + (−68.2 − 16.1i)17-s − 17.1·18-s − 28·19-s + ⋯
L(s)  = 1  − 0.838·2-s + 0.855i·3-s − 0.296·4-s − 0.717i·6-s + 0.809i·7-s + 1.08·8-s + 0.267·9-s − 0.853i·11-s − 0.253i·12-s + 0.111·13-s − 0.678i·14-s − 0.615·16-s + (−0.973 − 0.230i)17-s − 0.224·18-s − 0.338·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ 0.973 + 0.230i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8565015738\)
\(L(\frac12)\) \(\approx\) \(0.8565015738\)
\(L(\frac{5}{2})\) 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 \( 1 + (68.2 + 16.1i)T \)
good2 \( 1 + 2.37T + 8T^{2} \)
3 \( 1 - 4.44iT - 27T^{2} \)
7 \( 1 - 14.9iT - 343T^{2} \)
11 \( 1 + 31.1iT - 1.33e3T^{2} \)
13 \( 1 - 5.21T + 2.19e3T^{2} \)
19 \( 1 + 28T + 6.85e3T^{2} \)
23 \( 1 + 167. iT - 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 - 50.5iT - 2.97e4T^{2} \)
37 \( 1 + 260. iT - 5.06e4T^{2} \)
41 \( 1 + 183. iT - 6.89e4T^{2} \)
43 \( 1 - 348.T + 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 - 408.T + 1.48e5T^{2} \)
59 \( 1 - 108.T + 2.05e5T^{2} \)
61 \( 1 - 123. iT - 2.26e5T^{2} \)
67 \( 1 - 243.T + 3.00e5T^{2} \)
71 \( 1 + 42.7iT - 3.57e5T^{2} \)
73 \( 1 - 875. iT - 3.89e5T^{2} \)
79 \( 1 + 750. iT - 4.93e5T^{2} \)
83 \( 1 - 472.T + 5.71e5T^{2} \)
89 \( 1 - 376.T + 7.04e5T^{2} \)
97 \( 1 + 303. iT - 9.12e5T^{2} \)
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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

−10.58461734724667212366941384996, −9.717841142813899046785990006814, −8.845552257676266702733787603568, −8.520910843649280271644611838517, −7.18825044105028643531564357532, −5.91349631960334261859507065705, −4.76131669647939993586251254271, −3.92968441240067408143694272524, −2.30069642419324726043719554300, −0.49373279879810354413492577740, 0.982651357100946440302400256715, 1.93160990651406560422944146320, 3.92970079689356618183035638163, 4.87281680621578449516138642663, 6.52203529913626501108262580854, 7.32353194724888162684135038299, 7.891243728892699093113477650752, 8.980823430606244750880512550757, 9.851854147377861943690216766324, 10.56203236690445785923292801738

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