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Properties

Label 2-2007-223.87-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.703 - 0.710i$
Analytic cond. $1.00162$
Root an. cond. $1.00081$
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  + (0.828 + 0.559i)4-s + (0.743 + 0.0632i)7-s + (0.316 + 1.21i)13-s + (0.372 + 0.927i)16-s + (−1.90 + 0.327i)19-s + (0.210 + 0.977i)25-s + (0.580 + 0.468i)28-s + (0.848 − 1.68i)31-s + (−0.472 − 0.766i)37-s + (0.415 − 1.92i)43-s + (−0.437 − 0.0750i)49-s + (−0.418 + 1.18i)52-s + (−0.0427 − 0.164i)61-s + (−0.210 + 0.977i)64-s + (−0.840 + 1.24i)67-s + ⋯
L(s)  = 1  + (0.828 + 0.559i)4-s + (0.743 + 0.0632i)7-s + (0.316 + 1.21i)13-s + (0.372 + 0.927i)16-s + (−1.90 + 0.327i)19-s + (0.210 + 0.977i)25-s + (0.580 + 0.468i)28-s + (0.848 − 1.68i)31-s + (−0.472 − 0.766i)37-s + (0.415 − 1.92i)43-s + (−0.437 − 0.0750i)49-s + (−0.418 + 1.18i)52-s + (−0.0427 − 0.164i)61-s + (−0.210 + 0.977i)64-s + (−0.840 + 1.24i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1648, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.493944218\)
\(L(\frac12)\) \(\approx\) \(1.493944218\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 + (0.210 + 0.977i)T \)
good2 \( 1 + (-0.828 - 0.559i)T^{2} \)
5 \( 1 + (-0.210 - 0.977i)T^{2} \)
7 \( 1 + (-0.743 - 0.0632i)T + (0.985 + 0.169i)T^{2} \)
11 \( 1 + (0.450 + 0.892i)T^{2} \)
13 \( 1 + (-0.316 - 1.21i)T + (-0.873 + 0.487i)T^{2} \)
17 \( 1 + (-0.450 - 0.892i)T^{2} \)
19 \( 1 + (1.90 - 0.327i)T + (0.942 - 0.333i)T^{2} \)
23 \( 1 + (-0.210 - 0.977i)T^{2} \)
29 \( 1 + (-0.721 + 0.691i)T^{2} \)
31 \( 1 + (-0.848 + 1.68i)T + (-0.594 - 0.803i)T^{2} \)
37 \( 1 + (0.472 + 0.766i)T + (-0.450 + 0.892i)T^{2} \)
41 \( 1 + (-0.967 - 0.251i)T^{2} \)
43 \( 1 + (-0.415 + 1.92i)T + (-0.911 - 0.411i)T^{2} \)
47 \( 1 + (0.942 + 0.333i)T^{2} \)
53 \( 1 + (-0.996 + 0.0848i)T^{2} \)
59 \( 1 + (0.127 + 0.991i)T^{2} \)
61 \( 1 + (0.0427 + 0.164i)T + (-0.873 + 0.487i)T^{2} \)
67 \( 1 + (0.840 - 1.24i)T + (-0.372 - 0.927i)T^{2} \)
71 \( 1 + (0.127 + 0.991i)T^{2} \)
73 \( 1 + (-1.56 - 0.552i)T + (0.778 + 0.628i)T^{2} \)
79 \( 1 + (-1.93 - 0.248i)T + (0.967 + 0.251i)T^{2} \)
83 \( 1 + (-0.911 + 0.411i)T^{2} \)
89 \( 1 + (0.210 - 0.977i)T^{2} \)
97 \( 1 + (1.24 + 1.54i)T + (-0.210 + 0.977i)T^{2} \)
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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

−9.267575355691058674054121686940, −8.541638111168162299423391974999, −7.935639253160987901635700494016, −7.04739294073181656956376458077, −6.44535990713029700519604198736, −5.57192118315658345437865806452, −4.33291652969825903162861095280, −3.78825109603843128252586467625, −2.37159980275501808591285259394, −1.76832145496435238904048163021, 1.16841068482769135495507879149, 2.29178783191600500399782247785, 3.19239846358026516278708386743, 4.57356660858592992527850176231, 5.18062661737138107519695693938, 6.35645217667418046311959964114, 6.57217077110409045329812993810, 7.920273967241137601547555221625, 8.203759469122092748610554137204, 9.240216525696572810340030435056

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