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Properties

Label 2-4140-15.8-c1-0-35
Degree $2$
Conductor $4140$
Sign $-0.285 + 0.958i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.210 + 2.22i)5-s + (−0.752 + 0.752i)7-s − 4.75i·11-s + (2.05 + 2.05i)13-s + (−0.677 − 0.677i)17-s − 5.92i·19-s + (0.707 − 0.707i)23-s + (−4.91 + 0.936i)25-s + 0.804·29-s − 4.39·31-s + (−1.83 − 1.51i)35-s + (−4.33 + 4.33i)37-s − 2.94i·41-s + (−7.65 − 7.65i)43-s + (−1.31 − 1.31i)47-s + ⋯
L(s)  = 1  + (0.0941 + 0.995i)5-s + (−0.284 + 0.284i)7-s − 1.43i·11-s + (0.571 + 0.571i)13-s + (−0.164 − 0.164i)17-s − 1.35i·19-s + (0.147 − 0.147i)23-s + (−0.982 + 0.187i)25-s + 0.149·29-s − 0.788·31-s + (−0.309 − 0.256i)35-s + (−0.713 + 0.713i)37-s − 0.459i·41-s + (−1.16 − 1.16i)43-s + (−0.192 − 0.192i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7953675125\)
\(L(\frac12)\) \(\approx\) \(0.7953675125\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.210 - 2.22i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (0.752 - 0.752i)T - 7iT^{2} \)
11 \( 1 + 4.75iT - 11T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (0.677 + 0.677i)T + 17iT^{2} \)
19 \( 1 + 5.92iT - 19T^{2} \)
29 \( 1 - 0.804T + 29T^{2} \)
31 \( 1 + 4.39T + 31T^{2} \)
37 \( 1 + (4.33 - 4.33i)T - 37iT^{2} \)
41 \( 1 + 2.94iT - 41T^{2} \)
43 \( 1 + (7.65 + 7.65i)T + 43iT^{2} \)
47 \( 1 + (1.31 + 1.31i)T + 47iT^{2} \)
53 \( 1 + (2.70 - 2.70i)T - 53iT^{2} \)
59 \( 1 - 1.18T + 59T^{2} \)
61 \( 1 + 1.39T + 61T^{2} \)
67 \( 1 + (5.77 - 5.77i)T - 67iT^{2} \)
71 \( 1 - 0.358iT - 71T^{2} \)
73 \( 1 + (7.71 + 7.71i)T + 73iT^{2} \)
79 \( 1 + 16.1iT - 79T^{2} \)
83 \( 1 + (4.51 - 4.51i)T - 83iT^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + (9.86 - 9.86i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.353320863076279442440583860580, −7.26513048340562206064072845250, −6.74209342667764095477629753007, −6.07834333359552730977115441675, −5.40601114826734524272499328588, −4.32043621202197180347299169960, −3.30681715050874814129912199676, −2.90336221755119998900877782672, −1.74432829737077067774565932255, −0.22286235303326372555225046058, 1.28451038128401225056088509055, 2.00873623146496912510346748291, 3.39340135763569991221512748970, 4.11072660703334511468840278864, 4.90525179199765993460749731611, 5.59849123518458618081025727711, 6.40130992413937282628516335094, 7.26207555472196723378219501302, 7.976137276477356661303750768516, 8.550822320750284469271092306820

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