Properties

Label 2-4140-15.8-c1-0-38
Degree $2$
Conductor $4140$
Sign $-0.981 + 0.189i$
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  + (−1.77 − 1.36i)5-s + (3.33 − 3.33i)7-s + 1.66i·11-s + (−2.55 − 2.55i)13-s + (−1.21 − 1.21i)17-s − 2.64i·19-s + (−0.707 + 0.707i)23-s + (1.27 + 4.83i)25-s − 1.63·29-s + 1.29·31-s + (−10.4 + 1.35i)35-s + (5.51 − 5.51i)37-s − 10.2i·41-s + (−3.39 − 3.39i)43-s + (4.00 + 4.00i)47-s + ⋯
L(s)  = 1  + (−0.792 − 0.610i)5-s + (1.25 − 1.25i)7-s + 0.503i·11-s + (−0.709 − 0.709i)13-s + (−0.294 − 0.294i)17-s − 0.606i·19-s + (−0.147 + 0.147i)23-s + (0.255 + 0.966i)25-s − 0.302·29-s + 0.232·31-s + (−1.76 + 0.229i)35-s + (0.906 − 0.906i)37-s − 1.59i·41-s + (−0.518 − 0.518i)43-s + (0.583 + 0.583i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.981 + 0.189i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.031472485\)
\(L(\frac12)\) \(\approx\) \(1.031472485\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 + 1.36i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-3.33 + 3.33i)T - 7iT^{2} \)
11 \( 1 - 1.66iT - 11T^{2} \)
13 \( 1 + (2.55 + 2.55i)T + 13iT^{2} \)
17 \( 1 + (1.21 + 1.21i)T + 17iT^{2} \)
19 \( 1 + 2.64iT - 19T^{2} \)
29 \( 1 + 1.63T + 29T^{2} \)
31 \( 1 - 1.29T + 31T^{2} \)
37 \( 1 + (-5.51 + 5.51i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (3.39 + 3.39i)T + 43iT^{2} \)
47 \( 1 + (-4.00 - 4.00i)T + 47iT^{2} \)
53 \( 1 + (2.13 - 2.13i)T - 53iT^{2} \)
59 \( 1 + 4.14T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + (-3.61 + 3.61i)T - 67iT^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (-3.28 - 3.28i)T + 73iT^{2} \)
79 \( 1 + 1.68iT - 79T^{2} \)
83 \( 1 + (7.92 - 7.92i)T - 83iT^{2} \)
89 \( 1 + 2.20T + 89T^{2} \)
97 \( 1 + (7.00 - 7.00i)T - 97iT^{2} \)
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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

−7.905491094223631354032239192640, −7.43210521603747386473240056103, −7.02118542721739031875788135303, −5.62083542304589476800204872413, −4.85551205464572122405624297870, −4.39816878399290085738387994083, −3.68218005407057761405020996612, −2.41206316283838114312834388470, −1.26560215438751436302531406200, −0.30474939579043168809457701288, 1.54576448376785254723277246101, 2.45579206272878171588062736192, 3.25370469382852128105101475715, 4.42317806142477513994928460254, 4.84063844056387383553574106169, 5.92495871547110220858705196850, 6.46176116071001962921802176749, 7.51811781816113334751091550676, 8.058124992726074145492421747809, 8.557254545483622713347935667852

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