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Properties

Label 1-4008-4008.995-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.963 - 0.268i$
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  + (0.243 + 0.969i)5-s + (−0.726 + 0.686i)7-s + (−0.822 + 0.569i)11-s + (0.206 + 0.978i)13-s + (−0.455 − 0.890i)17-s + (0.132 + 0.991i)19-s + (0.700 − 0.713i)23-s + (−0.881 + 0.472i)25-s + (−0.700 − 0.713i)29-s + (−0.988 + 0.150i)31-s + (−0.843 − 0.537i)35-s + (−0.644 + 0.764i)37-s + (0.954 − 0.298i)41-s + (−0.553 + 0.832i)43-s + (0.489 − 0.872i)47-s + ⋯
L(s)  = 1  + (0.243 + 0.969i)5-s + (−0.726 + 0.686i)7-s + (−0.822 + 0.569i)11-s + (0.206 + 0.978i)13-s + (−0.455 − 0.890i)17-s + (0.132 + 0.991i)19-s + (0.700 − 0.713i)23-s + (−0.881 + 0.472i)25-s + (−0.700 − 0.713i)29-s + (−0.988 + 0.150i)31-s + (−0.843 − 0.537i)35-s + (−0.644 + 0.764i)37-s + (0.954 − 0.298i)41-s + (−0.553 + 0.832i)43-s + (0.489 − 0.872i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (995, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.963 - 0.268i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6754026818 - 0.09229069842i\)
\(L(\frac12)\) \(\approx\) \(0.6754026818 - 0.09229069842i\)
\(L(1)\) \(\approx\) \(0.7824902253 + 0.2836877037i\)
\(L(1)\) \(\approx\) \(0.7824902253 + 0.2836877037i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.243 + 0.969i)T \)
7 \( 1 + (-0.726 + 0.686i)T \)
11 \( 1 + (-0.822 + 0.569i)T \)
13 \( 1 + (0.206 + 0.978i)T \)
17 \( 1 + (-0.455 - 0.890i)T \)
19 \( 1 + (0.132 + 0.991i)T \)
23 \( 1 + (0.700 - 0.713i)T \)
29 \( 1 + (-0.700 - 0.713i)T \)
31 \( 1 + (-0.988 + 0.150i)T \)
37 \( 1 + (-0.644 + 0.764i)T \)
41 \( 1 + (0.954 - 0.298i)T \)
43 \( 1 + (-0.553 + 0.832i)T \)
47 \( 1 + (0.489 - 0.872i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (-0.455 + 0.890i)T \)
61 \( 1 + (-0.862 + 0.505i)T \)
67 \( 1 + (0.243 - 0.969i)T \)
71 \( 1 + (-0.997 + 0.0756i)T \)
73 \( 1 + (0.0944 - 0.995i)T \)
79 \( 1 + (0.280 + 0.959i)T \)
83 \( 1 + (-0.914 - 0.404i)T \)
89 \( 1 + (0.982 - 0.188i)T \)
97 \( 1 + (0.988 + 0.150i)T \)
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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

−18.261416723832756048975838983932, −17.458161751394122300673053998763, −17.08984652934250200237838244426, −16.22142966692969668426640738895, −15.77744407305279837869977605905, −15.13030826247363766914809028775, −14.05915389737778492151040809763, −13.26170908333612175908768934502, −13.006660105008380725554512036468, −12.528704232163290726438458163957, −11.27878159937747086332167148666, −10.730872819162183188466541423196, −10.11996347094904445644293029364, −9.10538435320429661275220308133, −8.83336492030468073167619989915, −7.77292874330963227376690882711, −7.2893632934579789718156850679, −6.22044680619409463936452843482, −5.54350525354062229639084569530, −4.9913558501418686631868103461, −3.94867315470036652042176777002, −3.34349861987170661868681504736, −2.41750591781912418405378630209, −1.32746459002457258141725140435, −0.54989524435343558615583969132, 0.155730328808474330801014982168, 1.718855365221666364328028464958, 2.38577660072438143283082567715, 3.01952000067320756683217272788, 3.85824971224766302174251599397, 4.817395399656992675583039184601, 5.69444297162405338737153350087, 6.30618425473737553686888766524, 7.07512483548345204202478195463, 7.54775935508474517499183458340, 8.701316255511740433913886708357, 9.321000289089525792771290891363, 9.99621935995944536013350740327, 10.61156819854158853290746305546, 11.4358520794951914212483011794, 12.05109565478375012841183677129, 12.85590600058430100777853512438, 13.536593885276012277345181001102, 14.173398344258066344376676170542, 15.073749932322764786526643922825, 15.34451053915755897092662112998, 16.34052978719256408530177121232, 16.718723009299744010378013141348, 17.90641244165326336751281655378, 18.39017723040038859535862794094

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