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Properties

Label 2-4140-15.8-c1-0-8
Degree $2$
Conductor $4140$
Sign $-0.337 - 0.941i$
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  + (−1.07 + 1.95i)5-s + (2.36 − 2.36i)7-s + 1.43i·11-s + (−4.27 − 4.27i)13-s + (0.739 + 0.739i)17-s + 3.16i·19-s + (0.707 − 0.707i)23-s + (−2.67 − 4.22i)25-s − 8.01·29-s + 7.15·31-s + (2.08 + 7.16i)35-s + (−6.14 + 6.14i)37-s + 10.2i·41-s + (−1.42 − 1.42i)43-s + (5.93 + 5.93i)47-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)5-s + (0.892 − 0.892i)7-s + 0.432i·11-s + (−1.18 − 1.18i)13-s + (0.179 + 0.179i)17-s + 0.726i·19-s + (0.147 − 0.147i)23-s + (−0.535 − 0.844i)25-s − 1.48·29-s + 1.28·31-s + (0.351 + 1.21i)35-s + (−1.01 + 1.01i)37-s + 1.60i·41-s + (−0.217 − 0.217i)43-s + (0.865 + 0.865i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.337 - 0.941i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066003614\)
\(L(\frac12)\) \(\approx\) \(1.066003614\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 - 1.95i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-2.36 + 2.36i)T - 7iT^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 + (4.27 + 4.27i)T + 13iT^{2} \)
17 \( 1 + (-0.739 - 0.739i)T + 17iT^{2} \)
19 \( 1 - 3.16iT - 19T^{2} \)
29 \( 1 + 8.01T + 29T^{2} \)
31 \( 1 - 7.15T + 31T^{2} \)
37 \( 1 + (6.14 - 6.14i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (1.42 + 1.42i)T + 43iT^{2} \)
47 \( 1 + (-5.93 - 5.93i)T + 47iT^{2} \)
53 \( 1 + (4.27 - 4.27i)T - 53iT^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 + (2.37 - 2.37i)T - 67iT^{2} \)
71 \( 1 + 2.71iT - 71T^{2} \)
73 \( 1 + (9.82 + 9.82i)T + 73iT^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + (5.86 - 5.86i)T - 83iT^{2} \)
89 \( 1 - 5.70T + 89T^{2} \)
97 \( 1 + (9.11 - 9.11i)T - 97iT^{2} \)
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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

−8.216334950753885435849830322255, −7.891466784625547336364829342020, −7.31257619866750115558362734455, −6.64969279517422430844056856207, −5.62553646088175816112388114814, −4.81207578607539038278121517636, −4.10335427425947181298825273993, −3.22676442632314512395007352349, −2.36898865238737761472699133593, −1.14603150641216838485523578199, 0.31927028863870041965726210750, 1.74439945495774566942906684297, 2.42968963158498167682783526048, 3.73799238137365935893849818913, 4.52016932075721916314098913355, 5.23677058720628311623674704171, 5.64910039377653221798652435239, 6.99832635718267718599406011180, 7.40433342098221166401660153407, 8.430191159313587181553492024728

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