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Properties

Label 1-4008-4008.371-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.335 + 0.942i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (−0.672 + 0.739i)5-s + (0.644 − 0.764i)7-s + (0.243 − 0.969i)11-s + (−0.584 + 0.811i)13-s + (0.988 − 0.150i)17-s + (−0.387 + 0.922i)19-s + (−0.726 + 0.686i)23-s + (−0.0944 − 0.995i)25-s + (0.726 + 0.686i)29-s + (−0.898 − 0.438i)31-s + (0.132 + 0.991i)35-s + (0.862 − 0.505i)37-s + (0.614 + 0.788i)41-s + (0.982 − 0.188i)43-s + (−0.999 − 0.0378i)47-s + ⋯
L(s)  = 1  + (−0.672 + 0.739i)5-s + (0.644 − 0.764i)7-s + (0.243 − 0.969i)11-s + (−0.584 + 0.811i)13-s + (0.988 − 0.150i)17-s + (−0.387 + 0.922i)19-s + (−0.726 + 0.686i)23-s + (−0.0944 − 0.995i)25-s + (0.726 + 0.686i)29-s + (−0.898 − 0.438i)31-s + (0.132 + 0.991i)35-s + (0.862 − 0.505i)37-s + (0.614 + 0.788i)41-s + (0.982 − 0.188i)43-s + (−0.999 − 0.0378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.335 + 0.942i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.335 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394166366 + 0.9839476926i\)
\(L(\frac12)\) \(\approx\) \(1.394166366 + 0.9839476926i\)
\(L(1)\) \(\approx\) \(0.9933539048 + 0.09359512865i\)
\(L(1)\) \(\approx\) \(0.9933539048 + 0.09359512865i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.672 + 0.739i)T \)
7 \( 1 + (0.644 - 0.764i)T \)
11 \( 1 + (0.243 - 0.969i)T \)
13 \( 1 + (-0.584 + 0.811i)T \)
17 \( 1 + (0.988 - 0.150i)T \)
19 \( 1 + (-0.387 + 0.922i)T \)
23 \( 1 + (-0.726 + 0.686i)T \)
29 \( 1 + (0.726 + 0.686i)T \)
31 \( 1 + (-0.898 - 0.438i)T \)
37 \( 1 + (0.862 - 0.505i)T \)
41 \( 1 + (0.614 + 0.788i)T \)
43 \( 1 + (0.982 - 0.188i)T \)
47 \( 1 + (-0.999 - 0.0378i)T \)
53 \( 1 + (0.942 + 0.334i)T \)
59 \( 1 + (0.988 + 0.150i)T \)
61 \( 1 + (0.0189 - 0.999i)T \)
67 \( 1 + (-0.672 - 0.739i)T \)
71 \( 1 + (-0.974 - 0.225i)T \)
73 \( 1 + (-0.280 - 0.959i)T \)
79 \( 1 + (-0.752 + 0.658i)T \)
83 \( 1 + (-0.316 + 0.948i)T \)
89 \( 1 + (0.843 + 0.537i)T \)
97 \( 1 + (0.898 - 0.438i)T \)
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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

−17.93576070795625545374653023630, −17.64327925674099282831589267854, −16.84873463643701653978380306279, −16.03152650268128491989616760712, −15.48884739648286621295456157216, −14.67576450763685912194947147041, −14.49121652742665207765339643964, −13.03457937147114190502660092539, −12.73076755477597683758438730097, −11.8766145569237127834542390775, −11.67084848378494996729470820186, −10.51225309678380590976011726246, −9.88045253806054342354598468434, −8.988913049928573504936137517, −8.44872633479722869528460429751, −7.7015166177493806162458405694, −7.2063273778142008846229369660, −6.00631261922763480720205616652, −5.33270369232294820649451895912, −4.60811621931748482627858313070, −4.116915712147963056277777623658, −2.91648478982072888839078787221, −2.206988423232332502572966464288, −1.23776994265672142837659135341, −0.35085756764911413361867535116, 0.69812107669990265419696164699, 1.58815110374204374417702593050, 2.577844234102099521656279586542, 3.60187632838141619169201982329, 3.93812881038663584435750531514, 4.8410168637722333795005815946, 5.84228943565968507956304880441, 6.50094625806000826647524980673, 7.53495354288091001723169969676, 7.68674515729610578583251277576, 8.5637147928851992318238522467, 9.521458270422519151274860115390, 10.30063165042131650312693185419, 10.893762550177182110925057384596, 11.60317004958188243935059399965, 11.999100530538363091137399682417, 12.99534092341343187862612810863, 13.97640992226593867961789989974, 14.45432884064401026918732520343, 14.67590737869587391508650956722, 15.86519295612171964864990371821, 16.50980537744114247474022061776, 16.85529589368349458480780261083, 17.96022034945867035254363919740, 18.3786282015091676492788139010

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