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Properties

Label 2-2005-2005.1603-c0-0-3
Degree $2$
Conductor $2005$
Sign $0.850 + 0.525i$
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 − i)2-s − i·4-s + 5-s + (−1 + i)7-s + i·9-s + (1 − i)10-s + 2i·14-s + 16-s + (1 + i)18-s − i·20-s + 25-s + (1 + i)28-s − 2i·29-s + (1 − i)32-s + (−1 + i)35-s + 36-s + ⋯
L(s)  = 1  + (1 − i)2-s − i·4-s + 5-s + (−1 + i)7-s + i·9-s + (1 − i)10-s + 2i·14-s + 16-s + (1 + i)18-s − i·20-s + 25-s + (1 + i)28-s − 2i·29-s + (1 − i)32-s + (−1 + i)35-s + 36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2005\)    =    \(5 \cdot 401\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.00062\)
Root analytic conductor: \(1.00031\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2005} (1603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2005,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.143417792\)
\(L(\frac12)\) \(\approx\) \(2.143417792\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.643087538873229781352202371049, −8.667637977783349476637630414117, −7.78353551958036844312843160141, −6.51510200986283288967670550136, −5.82117491680344010526795957342, −5.22585919524142474087844407206, −4.36086444443641520119793377363, −3.16664143017195651785313608554, −2.49167989300290052548453385176, −1.81798979340279615411955186206, 1.29255610354854101434201687764, 3.12026842389227566809209836210, 3.68191271787778485252461221164, 4.73754542577102845056180279032, 5.52130334437881394348948072526, 6.45189275431516334361112919265, 6.67178073060197634594904605399, 7.37957451693754690946929877592, 8.589029666025352436802912103825, 9.464182653012213707342728886500

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