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Properties

Label 1-3360-3360.1763-r0-0-0
Degree $1$
Conductor $3360$
Sign $-0.0354 + 0.999i$
Analytic cond. $15.6037$
Root an. cond. $15.6037$
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  + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (0.707 + 0.707i)19-s − 23-s + (0.707 + 0.707i)29-s + 31-s + (0.707 + 0.707i)37-s + i·41-s + (−0.707 − 0.707i)43-s − i·47-s + (−0.707 − 0.707i)53-s + (−0.707 + 0.707i)59-s + (−0.707 − 0.707i)61-s + (−0.707 + 0.707i)67-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (0.707 + 0.707i)19-s − 23-s + (0.707 + 0.707i)29-s + 31-s + (0.707 + 0.707i)37-s + i·41-s + (−0.707 − 0.707i)43-s − i·47-s + (−0.707 − 0.707i)53-s + (−0.707 + 0.707i)59-s + (−0.707 − 0.707i)61-s + (−0.707 + 0.707i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0354 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0354 + 0.999i$
Analytic conductor: \(15.6037\)
Root analytic conductor: \(15.6037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3360,\ (0:\ ),\ -0.0354 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9167861033 + 0.9498992196i\)
\(L(\frac12)\) \(\approx\) \(0.9167861033 + 0.9498992196i\)
\(L(1)\) \(\approx\) \(1.012215913 + 0.1636891827i\)
\(L(1)\) \(\approx\) \(1.012215913 + 0.1636891827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 - T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + iT \)
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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

−18.57853624559316273440035631060, −17.796174725017438572828004073729, −17.46545842718440086129018063627, −16.634786394775821551838196514535, −15.683072444024508417641285992051, −15.418864327809686875174675905714, −14.31005355100599063562171450773, −13.99746413217670412315523112218, −13.05641195107467418003918031770, −12.24179880107187447301929215339, −11.8063150276680659779879096120, −11.01032950410579597883290306967, −9.939997024204965751351763843441, −9.680060571878344404829679229369, −8.8232461543066383124500946097, −7.80168277996823877331660812224, −7.357993038516673389742872930842, −6.48421536920427621101756308080, −5.7177220739040582735961489928, −4.73356780985274934674131642814, −4.32890862578859040098449982247, −3.08649644841005488732915121691, −2.54097397271025733481902400780, −1.46677069359279962628365061530, −0.41181202158876441218624123099, 1.13032353106527905072921129465, 1.85004516363178059270508570776, 2.93174770746317624006347457991, 3.71282214697465296794901066285, 4.44980958970209526489991375536, 5.3119768235793026320688357073, 6.28609751677768968647023064196, 6.6226426445775120832915809122, 7.781278758117651627193101268632, 8.307840813039220374950779460697, 9.12660720963553785526602200148, 9.91826557629556208395615580592, 10.44949512522751104896296126500, 11.542983278828767362650124340032, 11.907851577888858700649010129609, 12.63321302602724147878588090471, 13.65438787457697738427721829877, 14.104643862405702801739004856184, 14.787969529835776746106418908296, 15.5274206751095309902367505916, 16.57984139954902905909749957170, 16.66649882682570739607341522513, 17.619210313927611446561110160759, 18.36609929935823055950789629613, 19.040489598272001943442931436460

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