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Properties

Label 1-3004-3004.671-r0-0-0
Degree $1$
Conductor $3004$
Sign $-0.984 - 0.175i$
Analytic cond. $13.9505$
Root an. cond. $13.9505$
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  + (0.309 + 0.951i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + 11-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (0.309 + 0.951i)5-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)9-s + 11-s + (−0.809 − 0.587i)13-s + (−0.809 + 0.587i)15-s + (−0.309 + 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)23-s + (−0.809 + 0.587i)25-s + (−0.809 − 0.587i)27-s + (0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3004\)    =    \(2^{2} \cdot 751\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(13.9505\)
Root analytic conductor: \(13.9505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3004} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3004,\ (0:\ ),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1514154275 + 1.708170481i\)
\(L(\frac12)\) \(\approx\) \(-0.1514154275 + 1.708170481i\)
\(L(1)\) \(\approx\) \(0.9002786215 + 0.8136864558i\)
\(L(1)\) \(\approx\) \(0.9002786215 + 0.8136864558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−18.77860348021087986351827019408, −17.70255873659377662232161343149, −17.483353365728492612778333077723, −16.73898164819053482673588791685, −16.15861733914610059915766948862, −14.96848979280091170882458758307, −14.26100511566581888795110397560, −13.62044462276245091071255128349, −13.33297155698584811660484007469, −12.23356982030705561538892615653, −11.85073335776930134661600052616, −11.120604251514263693168586693868, −9.937676783510183344713768650, −9.10590379932719583097405481348, −8.87930921425584413005746547588, −7.708623759158792224252214542955, −7.12148066381867545558880293871, −6.67152640784505011230447591852, −5.45207199574168103839049382659, −4.84066545948238189217227065254, −3.95212459823676061241048160024, −3.004888526208066410589569603377, −1.913795512723744529503185836687, −1.29424910099381325050650926904, −0.48434814998100535195238292085, 1.57874762009434103543947645411, 2.44335756611108010563966420843, 3.15324635203030411337668326409, 3.81073298368474393891853960460, 4.89285903704704337426975837168, 5.49024202612512581325982196209, 6.3172045718785140446331489548, 7.09129683062643929850216502801, 8.20772295643662277099409602377, 8.68454467821229724073988574053, 9.61460880734631996521753755784, 10.09360423859577822381176809515, 10.80535561750563064140480263216, 11.6100248752716505371716635168, 12.1544779860000565935686136254, 13.22651323938167014801154230030, 14.17329508285072092893945235069, 14.71139869032022257431624379863, 15.03005329504531327648709111127, 15.73089712303682710244503504623, 16.67895926359281061343504267827, 17.34856264722769884265973522829, 17.94275640875079174934537412362, 18.852786816587639888148818449110, 19.42242981680581203974971348516

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