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Properties

Label 2-425-85.4-c1-0-5
Degree $2$
Conductor $425$
Sign $0.564 + 0.825i$
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  − 1.21·2-s + (−2.23 − 2.23i)3-s − 0.520·4-s + (2.72 + 2.72i)6-s + (−0.679 + 0.679i)7-s + 3.06·8-s + 7.02i·9-s + (2.22 + 2.22i)11-s + (1.16 + 1.16i)12-s + 2.02i·13-s + (0.827 − 0.827i)14-s − 2.68·16-s + (2.07 − 3.56i)17-s − 8.54i·18-s − 5.28i·19-s + ⋯
L(s)  = 1  − 0.860·2-s + (−1.29 − 1.29i)3-s − 0.260·4-s + (1.11 + 1.11i)6-s + (−0.256 + 0.256i)7-s + 1.08·8-s + 2.34i·9-s + (0.669 + 0.669i)11-s + (0.336 + 0.336i)12-s + 0.561i·13-s + (0.221 − 0.221i)14-s − 0.672·16-s + (0.503 − 0.864i)17-s − 2.01i·18-s − 1.21i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.564 + 0.825i$
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: \((2,\ 425,\ (\ :1/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401127 - 0.211733i\)
\(L(\frac12)\) \(\approx\) \(0.401127 - 0.211733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 + 3.56i)T \)
good2 \( 1 + 1.21T + 2T^{2} \)
3 \( 1 + (2.23 + 2.23i)T + 3iT^{2} \)
7 \( 1 + (0.679 - 0.679i)T - 7iT^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
13 \( 1 - 2.02iT - 13T^{2} \)
19 \( 1 + 5.28iT - 19T^{2} \)
23 \( 1 + (6.01 - 6.01i)T - 23iT^{2} \)
29 \( 1 + (-0.857 + 0.857i)T - 29iT^{2} \)
31 \( 1 + (-3.97 + 3.97i)T - 31iT^{2} \)
37 \( 1 + (-5.84 - 5.84i)T + 37iT^{2} \)
41 \( 1 + (1.04 + 1.04i)T + 41iT^{2} \)
43 \( 1 - 7.01T + 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 - 5.24T + 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 + (-2.70 - 2.70i)T + 61iT^{2} \)
67 \( 1 - 2.37iT - 67T^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 + (-5.51 - 5.51i)T + 73iT^{2} \)
79 \( 1 + (-4.74 - 4.74i)T + 79iT^{2} \)
83 \( 1 - 0.171T + 83T^{2} \)
89 \( 1 + 1.32T + 89T^{2} \)
97 \( 1 + (1.33 + 1.33i)T + 97iT^{2} \)
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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

−11.31095567578344952660711356979, −9.997318007073115199336794291048, −9.367170984332468593720853867386, −8.096627566742903106725562401401, −7.28924699606629992974089467739, −6.59978991715011192285217375435, −5.48475760931231231201584621079, −4.44085179156479033080725969491, −2.05209085989288399509527379781, −0.75555773254394704378902217673, 0.821562154540774647364822320444, 3.72844424417423250187669153902, 4.39754983470426262002684093203, 5.68446300726818969773868059476, 6.33829078487114492843962242509, 7.917616023563246987281411825042, 8.838559803524822770364303280112, 9.786787447725747587478663728173, 10.38709540108969963309839691362, 10.80773339914138762663958305524

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