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Properties

Label 2-425-17.9-c1-0-4
Degree $2$
Conductor $425$
Sign $0.995 - 0.0932i$
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  + (1.09 − 1.09i)2-s + (−2.77 − 1.15i)3-s − 0.419i·4-s + (−4.32 + 1.79i)6-s + (1.32 + 3.19i)7-s + (1.73 + 1.73i)8-s + (4.27 + 4.27i)9-s + (−3.92 + 1.62i)11-s + (−0.483 + 1.16i)12-s + 0.127i·13-s + (4.96 + 2.05i)14-s + 4.66·16-s + (4.11 − 0.193i)17-s + 9.40·18-s + (1.81 − 1.81i)19-s + ⋯
L(s)  = 1  + (0.777 − 0.777i)2-s + (−1.60 − 0.664i)3-s − 0.209i·4-s + (−1.76 + 0.731i)6-s + (0.499 + 1.20i)7-s + (0.614 + 0.614i)8-s + (1.42 + 1.42i)9-s + (−1.18 + 0.489i)11-s + (−0.139 + 0.336i)12-s + 0.0353i·13-s + (1.32 + 0.549i)14-s + 1.16·16-s + (0.998 − 0.0468i)17-s + 2.21·18-s + (0.417 − 0.417i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.995 - 0.0932i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.995 - 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20385 + 0.0562755i\)
\(L(\frac12)\) \(\approx\) \(1.20385 + 0.0562755i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.11 + 0.193i)T \)
good2 \( 1 + (-1.09 + 1.09i)T - 2iT^{2} \)
3 \( 1 + (2.77 + 1.15i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.32 - 3.19i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (3.92 - 1.62i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.127iT - 13T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 + (3.00 - 1.24i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.87 - 4.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-4.95 - 2.05i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.63 - 0.677i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.85 - 9.29i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.79 + 1.79i)T + 43iT^{2} \)
47 \( 1 - 4.59iT - 47T^{2} \)
53 \( 1 + (1.15 - 1.15i)T - 53iT^{2} \)
59 \( 1 + (4.34 + 4.34i)T + 59iT^{2} \)
61 \( 1 + (1.54 + 3.73i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 - 6.88T + 67T^{2} \)
71 \( 1 + (6.66 + 2.76i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.59 + 13.5i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.75 - 1.97i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.2 - 10.2i)T - 83iT^{2} \)
89 \( 1 - 0.600iT - 89T^{2} \)
97 \( 1 + (-2.93 + 7.09i)T + (-68.5 - 68.5i)T^{2} \)
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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

−11.44971952620389337003675759359, −10.82131860114184795124331234783, −9.839659080187629224601340687244, −8.154501890104553171847268895593, −7.44000560504812601792274825655, −6.04911997622779461269710610886, −5.24461382265393835603237419230, −4.75545985596132018919377297920, −2.85998128938122672819820782373, −1.63624094296647318932086455311, 0.791856920807198456671094187236, 3.83224248747553335091928975278, 4.61086329166831663118854708101, 5.51185489174279175088460615606, 6.00834502636766505256026149445, 7.19726231620007273469876434629, 7.929199704415774439867356414224, 10.00859690040254040504358488583, 10.27042832548700614944361770467, 11.07456510179578133721561304093

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