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Properties

Label 2-425-5.4-c1-0-6
Degree $2$
Conductor $425$
Sign $0.447 + 0.894i$
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  − 2.60i·2-s + 1.18i·3-s − 4.77·4-s + 3.07·6-s + 3.53i·7-s + 7.21i·8-s + 1.60·9-s + 2.94·11-s − 5.64i·12-s − 4.01i·13-s + 9.20·14-s + 9.23·16-s − i·17-s − 4.17i·18-s + 6.97·19-s + ⋯
L(s)  = 1  − 1.84i·2-s + 0.682i·3-s − 2.38·4-s + 1.25·6-s + 1.33i·7-s + 2.55i·8-s + 0.534·9-s + 0.888·11-s − 1.62i·12-s − 1.11i·13-s + 2.45·14-s + 2.30·16-s − 0.242i·17-s − 0.982i·18-s + 1.60·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11133 - 0.686841i\)
\(L(\frac12)\) \(\approx\) \(1.11133 - 0.686841i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good2 \( 1 + 2.60iT - 2T^{2} \)
3 \( 1 - 1.18iT - 3T^{2} \)
7 \( 1 - 3.53iT - 7T^{2} \)
11 \( 1 - 2.94T + 11T^{2} \)
13 \( 1 + 4.01iT - 13T^{2} \)
19 \( 1 - 6.97T + 19T^{2} \)
23 \( 1 - 6.12iT - 23T^{2} \)
29 \( 1 + 5.30T + 29T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 - 3.43iT - 37T^{2} \)
41 \( 1 - 4.61T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 3.67iT - 47T^{2} \)
53 \( 1 - 6.77iT - 53T^{2} \)
59 \( 1 + 9.92T + 59T^{2} \)
61 \( 1 + 2.36T + 61T^{2} \)
67 \( 1 + 9.56iT - 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 2.00iT - 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 9.07iT - 83T^{2} \)
89 \( 1 + 2.63T + 89T^{2} \)
97 \( 1 - 5.86iT - 97T^{2} \)
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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

−11.09931396819239483260908206238, −10.12657452675738637520919582224, −9.420240838368438067114765426022, −9.022280041372009947997764724996, −7.68284614012553487898814891924, −5.70037057967726403412558192624, −4.89797794904517268884112418508, −3.67483966662507325573209796404, −2.88374759307038340463624944815, −1.40673212829639312385410396829, 1.08643592367414973503254490501, 3.96962045639960560314002283185, 4.62538315514944369514576233336, 6.11819981016611825560801935751, 6.87287151942463214187884599847, 7.31557843041070624485363852699, 8.158411995363067398596717412508, 9.298604174750196093874098908080, 9.990754210258589083741113749453, 11.43210721596858629466466408306

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