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Properties

Label 2-425-17.16-c1-0-15
Degree $2$
Conductor $425$
Sign $0.395 + 0.918i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.17·2-s + 2.21i·3-s + 2.70·4-s − 4.80i·6-s − 1.02i·7-s − 1.53·8-s − 1.90·9-s − 4.98i·11-s + 6.00i·12-s − 3.87·13-s + 2.21i·14-s − 2.07·16-s + (−1.63 − 3.78i)17-s + 4.14·18-s − 4.04·19-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.27i·3-s + 1.35·4-s − 1.96i·6-s − 0.385i·7-s − 0.544·8-s − 0.636·9-s − 1.50i·11-s + 1.73i·12-s − 1.07·13-s + 0.592i·14-s − 0.519·16-s + (−0.395 − 0.918i)17-s + 0.976·18-s − 0.929·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297255 - 0.195630i\)
\(L(\frac12)\) \(\approx\) \(0.297255 - 0.195630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 + 3.78i)T \)
good2 \( 1 + 2.17T + 2T^{2} \)
3 \( 1 - 2.21iT - 3T^{2} \)
7 \( 1 + 1.02iT - 7T^{2} \)
11 \( 1 + 4.98iT - 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 + 7.02iT - 23T^{2} \)
29 \( 1 - 0.644iT - 29T^{2} \)
31 \( 1 - 4.25iT - 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 5.82iT - 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 0.447T + 59T^{2} \)
61 \( 1 - 3.14iT - 61T^{2} \)
67 \( 1 - 5.07T + 67T^{2} \)
71 \( 1 + 3.41iT - 71T^{2} \)
73 \( 1 - 3.78iT - 73T^{2} \)
79 \( 1 - 0.376iT - 79T^{2} \)
83 \( 1 - 3.57T + 83T^{2} \)
89 \( 1 + 2.63T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−10.65474897129466243279251100133, −10.16010428258349583245033844577, −9.062679869640319872135077921679, −8.780715488712311792250486706288, −7.59071078824804189856085504124, −6.59818538607397303272394865941, −5.12812709971993763825023292629, −4.05394989446714981212819123688, −2.57184021284075513317618326857, −0.36731966096398744806428901814, 1.60232175629293469886525525029, 2.31235295365373587138834109000, 4.60195739282756116414221586722, 6.25585532301486514424437933036, 7.09751566484785483582400360297, 7.69638539384054837135982663333, 8.455309111822175732961099760329, 9.585712611053719742104421591502, 10.08543007720754962943263966324, 11.30460152926578676544708065696

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