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Properties

Label 1-2004-2004.815-r0-0-0
Degree $1$
Conductor $2004$
Sign $-0.674 - 0.737i$
Analytic cond. $9.30653$
Root an. cond. $9.30653$
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  + (−0.997 + 0.0756i)5-s + (0.881 + 0.472i)7-s + (−0.521 − 0.853i)11-s + (0.0567 − 0.998i)13-s + (0.644 − 0.764i)17-s + (0.316 − 0.948i)19-s + (0.351 − 0.936i)23-s + (0.988 − 0.150i)25-s + (−0.351 − 0.936i)29-s + (−0.862 + 0.505i)31-s + (−0.914 − 0.404i)35-s + (−0.0944 + 0.995i)37-s + (−0.489 + 0.872i)41-s + (−0.954 + 0.298i)43-s + (−0.843 + 0.537i)47-s + ⋯
L(s)  = 1  + (−0.997 + 0.0756i)5-s + (0.881 + 0.472i)7-s + (−0.521 − 0.853i)11-s + (0.0567 − 0.998i)13-s + (0.644 − 0.764i)17-s + (0.316 − 0.948i)19-s + (0.351 − 0.936i)23-s + (0.988 − 0.150i)25-s + (−0.351 − 0.936i)29-s + (−0.862 + 0.505i)31-s + (−0.914 − 0.404i)35-s + (−0.0944 + 0.995i)37-s + (−0.489 + 0.872i)41-s + (−0.954 + 0.298i)43-s + (−0.843 + 0.537i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2004\)    =    \(2^{2} \cdot 3 \cdot 167\)
Sign: $-0.674 - 0.737i$
Analytic conductor: \(9.30653\)
Root analytic conductor: \(9.30653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2004} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2004,\ (0:\ ),\ -0.674 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3241189967 - 0.7356483247i\)
\(L(\frac12)\) \(\approx\) \(0.3241189967 - 0.7356483247i\)
\(L(1)\) \(\approx\) \(0.8446286515 - 0.1746183482i\)
\(L(1)\) \(\approx\) \(0.8446286515 - 0.1746183482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.997 + 0.0756i)T \)
7 \( 1 + (0.881 + 0.472i)T \)
11 \( 1 + (-0.521 - 0.853i)T \)
13 \( 1 + (0.0567 - 0.998i)T \)
17 \( 1 + (0.644 - 0.764i)T \)
19 \( 1 + (0.316 - 0.948i)T \)
23 \( 1 + (0.351 - 0.936i)T \)
29 \( 1 + (-0.351 - 0.936i)T \)
31 \( 1 + (-0.862 + 0.505i)T \)
37 \( 1 + (-0.0944 + 0.995i)T \)
41 \( 1 + (-0.489 + 0.872i)T \)
43 \( 1 + (-0.954 + 0.298i)T \)
47 \( 1 + (-0.843 + 0.537i)T \)
53 \( 1 + (0.387 + 0.922i)T \)
59 \( 1 + (-0.644 - 0.764i)T \)
61 \( 1 + (0.280 + 0.959i)T \)
67 \( 1 + (-0.997 - 0.0756i)T \)
71 \( 1 + (-0.965 + 0.261i)T \)
73 \( 1 + (0.898 - 0.438i)T \)
79 \( 1 + (-0.206 - 0.978i)T \)
83 \( 1 + (-0.993 + 0.113i)T \)
89 \( 1 + (-0.614 - 0.788i)T \)
97 \( 1 + (0.862 + 0.505i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.149228230542069952904781189360, −19.611317744313641616157647660580, −18.6617053364903443782437603518, −18.21696057055824365184135135883, −17.176104005860550052631638782087, −16.611630144464842255469317817402, −15.8849387179700160529976207793, −14.86374412435426857417546088510, −14.66706258702990908082279849386, −13.66513384495662458469697301302, −12.718969805064345017526389094911, −12.05712157634819511146987331477, −11.377472882266994513108009108694, −10.68252666024074450577462799498, −9.88217302340945042825422109504, −8.87574414929311140416273880239, −8.083905881416206678472559799881, −7.429898578052376049782769055928, −6.94502118990692576027324770273, −5.53579769938268833368133330333, −4.90989254825689452223072394601, −3.92681461189898348689963296589, −3.5350525828775920828385966413, −1.97724937159594086120752667219, −1.372345653871659680093024813040, 0.290887653893789707225755196250, 1.34154764599069154563882579051, 2.91276060523199787563006152553, 3.06196582142043877529676927403, 4.47042846947864372932127940631, 5.05992296302051037934726622116, 5.844992948565846111079823377402, 6.98356148446443180937029303160, 7.82281643541472399424319496126, 8.265110781971603371782387429383, 8.99919883911379607135323599368, 10.141825321997258999735476108665, 11.001448658521668521014982838577, 11.45988416453248721886567598077, 12.16984012435002267766063220733, 13.04910238586583797113475376141, 13.80818568048068112513441660245, 14.83957659284671044715346073534, 15.172735357001378053187480076955, 16.02418846944648736039416427101, 16.60917309635234179279171899621, 17.62956063646572793582030906277, 18.453984503496721559218057752198, 18.71629361897927901434955862145, 19.75282407933882429050168139249

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