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Properties

Label 1-3021-3021.365-r1-0-0
Degree $1$
Conductor $3021$
Sign $0.0375 - 0.999i$
Analytic cond. $324.651$
Root an. cond. $324.651$
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  + (−0.964 + 0.265i)2-s + (0.859 − 0.511i)4-s + (0.956 + 0.291i)5-s + (−0.996 − 0.0804i)7-s + (−0.692 + 0.721i)8-s + (−0.999 − 0.0268i)10-s + (0.845 − 0.534i)11-s + (0.653 − 0.757i)13-s + (0.982 − 0.186i)14-s + (0.476 − 0.879i)16-s + (0.897 + 0.440i)17-s + (0.970 − 0.239i)20-s + (−0.673 + 0.739i)22-s + (−0.173 − 0.984i)23-s + (0.830 + 0.556i)25-s + (−0.428 + 0.903i)26-s + ⋯
L(s)  = 1  + (−0.964 + 0.265i)2-s + (0.859 − 0.511i)4-s + (0.956 + 0.291i)5-s + (−0.996 − 0.0804i)7-s + (−0.692 + 0.721i)8-s + (−0.999 − 0.0268i)10-s + (0.845 − 0.534i)11-s + (0.653 − 0.757i)13-s + (0.982 − 0.186i)14-s + (0.476 − 0.879i)16-s + (0.897 + 0.440i)17-s + (0.970 − 0.239i)20-s + (−0.673 + 0.739i)22-s + (−0.173 − 0.984i)23-s + (0.830 + 0.556i)25-s + (−0.428 + 0.903i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0375 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0375 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $0.0375 - 0.999i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ 0.0375 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9314224389 - 0.8970910525i\)
\(L(\frac12)\) \(\approx\) \(0.9314224389 - 0.8970910525i\)
\(L(1)\) \(\approx\) \(0.8116154193 + 0.02068228480i\)
\(L(1)\) \(\approx\) \(0.8116154193 + 0.02068228480i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.964 + 0.265i)T \)
5 \( 1 + (0.956 + 0.291i)T \)
7 \( 1 + (-0.996 - 0.0804i)T \)
11 \( 1 + (0.845 - 0.534i)T \)
13 \( 1 + (0.653 - 0.757i)T \)
17 \( 1 + (0.897 + 0.440i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.379 + 0.925i)T \)
31 \( 1 + (-0.845 - 0.534i)T \)
37 \( 1 + (-0.748 + 0.663i)T \)
41 \( 1 + (0.991 + 0.133i)T \)
43 \( 1 + (-0.783 + 0.621i)T \)
47 \( 1 + (-0.730 - 0.682i)T \)
59 \( 1 + (0.545 - 0.837i)T \)
61 \( 1 + (0.0670 - 0.997i)T \)
67 \( 1 + (-0.872 - 0.488i)T \)
71 \( 1 + (0.783 - 0.621i)T \)
73 \( 1 + (0.830 - 0.556i)T \)
79 \( 1 + (-0.815 - 0.579i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.994 - 0.107i)T \)
97 \( 1 + (0.730 - 0.682i)T \)
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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

−19.03064766828543734052122463730, −18.36475133850776553875580664554, −17.62813049101775940962389495857, −17.057800316009877334322019002882, −16.34115433824511031947747230362, −15.98386018812892085762799988991, −14.93035623083010264823325277185, −14.06619755428490688368101184688, −13.32645376028238071300228224847, −12.562978610238175520178097205175, −11.95067798147008470626599718903, −11.20298862019123638822403417995, −10.19555716386753444201214369619, −9.69439184230329517750808058759, −9.19580592033134931745865096542, −8.64975843186712860646308710795, −7.42798924046476344113740485596, −6.871562643154977587601862185588, −6.0965040008849598766597687102, −5.50001858067070443020452120446, −4.06641189582661211418747771238, −3.4054789584491021600630677173, −2.39991884881841877829343377266, −1.62571840973137731781685521355, −0.93877503528669769004141042812, 0.31407202563207179594997533284, 1.1937799904110343807402134871, 1.98780935235115031842265401749, 3.12248490525727832355229611438, 3.52142475701896773969468497512, 5.164692824742297673084005648429, 5.98113642033224133170312808544, 6.35011505934279781899570937218, 7.02949962278089648147040391912, 8.04977098077816447119594807447, 8.7849784554663338711455277946, 9.445660819258304416063108306008, 10.06663848139405097774106634470, 10.64780959139246822317377056868, 11.32885871685277897572152780552, 12.43063842248565735015004128868, 13.002227995773382712591826901321, 13.979897627767907700736133132206, 14.57127144639300797050521163023, 15.27609476990658724107279873537, 16.26882397604491208157265663006, 16.6724597661101179399239447783, 17.19137688417794873313869307884, 18.2347269216045939835545313661, 18.45968634260359602040801124749

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