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Properties

Label 2-4000-25.22-c0-0-2
Degree $2$
Conductor $4000$
Sign $0.0941 + 0.995i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)9-s + (−1.76 − 0.896i)13-s + (0.896 + 0.142i)17-s + (1.11 − 1.53i)29-s + (0.809 − 1.58i)37-s + (−0.363 − 1.11i)41-s − i·49-s + (−0.309 + 0.0489i)53-s + (0.363 − 1.11i)61-s + (−0.142 − 0.278i)73-s + (0.809 − 0.587i)81-s + (−1.80 − 0.587i)89-s + (0.278 + 1.76i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)9-s + (−1.76 − 0.896i)13-s + (0.896 + 0.142i)17-s + (1.11 − 1.53i)29-s + (0.809 − 1.58i)37-s + (−0.363 − 1.11i)41-s − i·49-s + (−0.309 + 0.0489i)53-s + (0.363 − 1.11i)61-s + (−0.142 − 0.278i)73-s + (0.809 − 0.587i)81-s + (−1.80 − 0.587i)89-s + (0.278 + 1.76i)97-s − 0.618·101-s + (−1.53 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.0941 + 0.995i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.0941 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8524204791\)
\(L(\frac12)\) \(\approx\) \(0.8524204791\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.951 - 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.76 + 0.896i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.896 - 0.142i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \)
41 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.142 + 0.278i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 - 0.309i)T^{2} \)
89 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−8.255250703109015014772193500180, −7.82624745702026319644781447997, −7.14191486344257992736525376526, −6.10486807889811791485659495572, −5.44108096429349854330290763003, −4.87006859435132667612283396699, −3.79701417062546298362384664642, −2.79372034512748355614855077828, −2.23057068338761059646618022487, −0.48071580671073338346511777072, 1.33089168249077969050786231808, 2.67601069272635916747390973873, 3.12291237010837719250098840176, 4.43311906660818571780168113229, 4.99205768008191945147831302792, 5.82775090448766577181918602843, 6.68804698416045192976207335732, 7.26112129689592523098074476606, 8.146103572199910578698194573441, 8.727632290513133356239263873421

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