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Properties

Label 2-425-85.4-c1-0-22
Degree $2$
Conductor $425$
Sign $0.807 + 0.590i$
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  + 2.51·2-s + (−0.887 − 0.887i)3-s + 4.31·4-s + (−2.22 − 2.22i)6-s + (1.14 − 1.14i)7-s + 5.80·8-s − 1.42i·9-s + (−2.32 − 2.32i)11-s + (−3.82 − 3.82i)12-s + 6.35i·13-s + (2.86 − 2.86i)14-s + 5.96·16-s + (4.05 + 0.768i)17-s − 3.58i·18-s + 0.747i·19-s + ⋯
L(s)  = 1  + 1.77·2-s + (−0.512 − 0.512i)3-s + 2.15·4-s + (−0.909 − 0.909i)6-s + (0.431 − 0.431i)7-s + 2.05·8-s − 0.475i·9-s + (−0.700 − 0.700i)11-s + (−1.10 − 1.10i)12-s + 1.76i·13-s + (0.766 − 0.766i)14-s + 1.49·16-s + (0.982 + 0.186i)17-s − 0.844i·18-s + 0.171i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.807 + 0.590i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.807 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.08107 - 1.00681i\)
\(L(\frac12)\) \(\approx\) \(3.08107 - 1.00681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.05 - 0.768i)T \)
good2 \( 1 - 2.51T + 2T^{2} \)
3 \( 1 + (0.887 + 0.887i)T + 3iT^{2} \)
7 \( 1 + (-1.14 + 1.14i)T - 7iT^{2} \)
11 \( 1 + (2.32 + 2.32i)T + 11iT^{2} \)
13 \( 1 - 6.35iT - 13T^{2} \)
19 \( 1 - 0.747iT - 19T^{2} \)
23 \( 1 + (0.101 - 0.101i)T - 23iT^{2} \)
29 \( 1 + (6.22 - 6.22i)T - 29iT^{2} \)
31 \( 1 + (-5.10 + 5.10i)T - 31iT^{2} \)
37 \( 1 + (-0.439 - 0.439i)T + 37iT^{2} \)
41 \( 1 + (4.49 + 4.49i)T + 41iT^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 - 11.9iT - 47T^{2} \)
53 \( 1 + 2.71T + 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 + (-0.328 - 0.328i)T + 61iT^{2} \)
67 \( 1 + 1.81iT - 67T^{2} \)
71 \( 1 + (3.01 - 3.01i)T - 71iT^{2} \)
73 \( 1 + (0.856 + 0.856i)T + 73iT^{2} \)
79 \( 1 + (-3.57 - 3.57i)T + 79iT^{2} \)
83 \( 1 - 3.58T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + (-4.92 - 4.92i)T + 97iT^{2} \)
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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

−11.42170791604722659081796458609, −10.83244244455478740008869691971, −9.388224016021485526227491786574, −7.86010574546180501835868367515, −6.91952327757268960866003161000, −6.16511927572550230683408203230, −5.31983591797557393276743784041, −4.25738228940925707605426503075, −3.27817981426568772526524014201, −1.67381586527416788209167785398, 2.31130793275224546725378773960, 3.42234583428524110389741527128, 4.77989227894048877000705840726, 5.26887451640006427872845566177, 5.92904357207324199660454562390, 7.37437389826641905022221539028, 8.149417700269247962196790669492, 10.06517665830239539459042643814, 10.54268220701193596530233645002, 11.59829724077205728400212890236

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