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Properties

Label 2-425-17.8-c1-0-10
Degree $2$
Conductor $425$
Sign $0.0250 - 0.999i$
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  + (0.187 − 0.187i)2-s + (−0.692 + 1.67i)3-s + 1.92i·4-s + (0.183 + 0.443i)6-s + (4.55 − 1.88i)7-s + (0.737 + 0.737i)8-s + (−0.193 − 0.193i)9-s + (1.50 + 3.62i)11-s + (−3.22 − 1.33i)12-s − 2.07i·13-s + (0.500 − 1.20i)14-s − 3.58·16-s + (−2.60 − 3.19i)17-s − 0.0724·18-s + (−2.08 + 2.08i)19-s + ⋯
L(s)  = 1  + (0.132 − 0.132i)2-s + (−0.399 + 0.965i)3-s + 0.964i·4-s + (0.0749 + 0.181i)6-s + (1.72 − 0.712i)7-s + (0.260 + 0.260i)8-s + (−0.0643 − 0.0643i)9-s + (0.452 + 1.09i)11-s + (−0.931 − 0.385i)12-s − 0.574i·13-s + (0.133 − 0.322i)14-s − 0.895·16-s + (−0.632 − 0.774i)17-s − 0.0170·18-s + (−0.478 + 0.478i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.0250 - 0.999i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.0250 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10544 + 1.07807i\)
\(L(\frac12)\) \(\approx\) \(1.10544 + 1.07807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 + 3.19i)T \)
good2 \( 1 + (-0.187 + 0.187i)T - 2iT^{2} \)
3 \( 1 + (0.692 - 1.67i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-4.55 + 1.88i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.50 - 3.62i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.07iT - 13T^{2} \)
19 \( 1 + (2.08 - 2.08i)T - 19iT^{2} \)
23 \( 1 + (-1.48 - 3.58i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.22 - 1.33i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.23 + 5.40i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.781 + 1.88i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (10.9 - 4.53i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 + 0.453iT - 47T^{2} \)
53 \( 1 + (-4.60 + 4.60i)T - 53iT^{2} \)
59 \( 1 + (-4.92 - 4.92i)T + 59iT^{2} \)
61 \( 1 + (0.0429 - 0.0177i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + (-5.85 + 14.1i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (8.12 + 3.36i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.991 + 2.39i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (1.09 - 1.09i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (9.83 + 4.07i)T + (68.5 + 68.5i)T^{2} \)
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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

−11.44016345711656714888404280616, −10.62027031251681841405235301094, −9.800739972818593589173618305651, −8.586163377109413434257079211037, −7.74277922172384623931289155151, −6.99004797054193826780589906634, −5.08925297812581618732046496507, −4.58869274237309410536994778005, −3.76965748351957442965673197422, −1.96576475243803318035014560653, 1.15420565659495745024230121073, 2.10451158432848796121913897361, 4.38378582792817956730805197148, 5.32438801964667477742857999877, 6.29961403213348803037858063107, 6.89926130467705294498535992114, 8.392424065980165109170605791492, 8.774640705488245095175418326427, 10.28307292456306936078407707610, 11.30806850017743872765667881709

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