Properties

Label 2-2005-2005.1202-c0-0-2
Degree $2$
Conductor $2005$
Sign $-0.237 - 0.971i$
Analytic cond. $1.00062$
Root an. cond. $1.00031$
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  + (1.26 + 1.26i)2-s + 2.17i·4-s + (0.309 − 0.951i)5-s + (1.39 + 1.39i)7-s + (−1.48 + 1.48i)8-s i·9-s + (1.58 − 0.809i)10-s − 1.17·11-s + 3.52i·14-s − 1.55·16-s + (1.26 − 1.26i)18-s + (2.06 + 0.672i)20-s + (−1.48 − 1.48i)22-s + (−0.809 − 0.587i)25-s + (−3.03 + 3.03i)28-s + 0.618i·29-s + ⋯
L(s)  = 1  + (1.26 + 1.26i)2-s + 2.17i·4-s + (0.309 − 0.951i)5-s + (1.39 + 1.39i)7-s + (−1.48 + 1.48i)8-s i·9-s + (1.58 − 0.809i)10-s − 1.17·11-s + 3.52i·14-s − 1.55·16-s + (1.26 − 1.26i)18-s + (2.06 + 0.672i)20-s + (−1.48 − 1.48i)22-s + (−0.809 − 0.587i)25-s + (−3.03 + 3.03i)28-s + 0.618i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $-0.237 - 0.971i$
Analytic conductor: \(1.00062\)
Root analytic conductor: \(1.00031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (1202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2005,\ (\ :0),\ -0.237 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.491820361\)
\(L(\frac12)\) \(\approx\) \(2.491820361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 + 0.951i)T \)
401 \( 1 - T \)
good2 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
3 \( 1 + iT^{2} \)
7 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
11 \( 1 + 1.17T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 0.618iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.90T + T^{2} \)
43 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
47 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + 1.17iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.016142504069642522258059663859, −8.557279670288952043749991563508, −8.013366851202917642052776495275, −7.10570851276592936763685471691, −6.06650009947125909204070365697, −5.38997178962007995603470554883, −5.11453788666098100197071892525, −4.26721897665098796983986463774, −3.08161237588720439488492947179, −1.90724786213332537769753829899, 1.49397951306707298342375207183, 2.30715847758054973690222186602, 3.17308709664276454036710089712, 4.22735888723288724380869978315, 4.85203195954914067115284641438, 5.47503551803549820358308711611, 6.57316502014356213471440733092, 7.68610582311065670807739122498, 7.990286224800823111931040551975, 9.700975749065559492443819996733

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