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Properties

Label 2-3060-15.2-c1-0-14
Degree $2$
Conductor $3060$
Sign $0.624 - 0.780i$
Analytic cond. $24.4342$
Root an. cond. $4.94309$
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  + (1.09 + 1.94i)5-s + (1.83 + 1.83i)7-s − 6.26i·11-s + (−0.715 + 0.715i)13-s + (−0.707 + 0.707i)17-s + 2.33i·19-s + (5.08 + 5.08i)23-s + (−2.59 + 4.27i)25-s + 8.45·29-s + 4.54·31-s + (−1.56 + 5.58i)35-s + (−6.15 − 6.15i)37-s − 7.65i·41-s + (−1.77 + 1.77i)43-s + (−3.43 + 3.43i)47-s + ⋯
L(s)  = 1  + (0.489 + 0.871i)5-s + (0.692 + 0.692i)7-s − 1.88i·11-s + (−0.198 + 0.198i)13-s + (−0.171 + 0.171i)17-s + 0.535i·19-s + (1.05 + 1.05i)23-s + (−0.519 + 0.854i)25-s + 1.56·29-s + 0.816·31-s + (−0.264 + 0.943i)35-s + (−1.01 − 1.01i)37-s − 1.19i·41-s + (−0.270 + 0.270i)43-s + (−0.500 + 0.500i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.624 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.256321507\)
\(L(\frac12)\) \(\approx\) \(2.256321507\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.94i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-1.83 - 1.83i)T + 7iT^{2} \)
11 \( 1 + 6.26iT - 11T^{2} \)
13 \( 1 + (0.715 - 0.715i)T - 13iT^{2} \)
19 \( 1 - 2.33iT - 19T^{2} \)
23 \( 1 + (-5.08 - 5.08i)T + 23iT^{2} \)
29 \( 1 - 8.45T + 29T^{2} \)
31 \( 1 - 4.54T + 31T^{2} \)
37 \( 1 + (6.15 + 6.15i)T + 37iT^{2} \)
41 \( 1 + 7.65iT - 41T^{2} \)
43 \( 1 + (1.77 - 1.77i)T - 43iT^{2} \)
47 \( 1 + (3.43 - 3.43i)T - 47iT^{2} \)
53 \( 1 + (-8.71 - 8.71i)T + 53iT^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 2.37T + 61T^{2} \)
67 \( 1 + (-4.00 - 4.00i)T + 67iT^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 + (3.27 - 3.27i)T - 73iT^{2} \)
79 \( 1 + 3.54iT - 79T^{2} \)
83 \( 1 + (2.67 + 2.67i)T + 83iT^{2} \)
89 \( 1 - 7.70T + 89T^{2} \)
97 \( 1 + (4.35 + 4.35i)T + 97iT^{2} \)
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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

−8.643940921215231989554470836225, −8.306230918123069300008585247267, −7.22402509506666684164608491291, −6.53398595471414766358627931301, −5.62023865169201484036764768595, −5.35639796606861647189920535631, −3.95518687559016641924573195358, −3.08506968093698740165761481342, −2.35811538490085840783779680380, −1.13464870146913584241519516390, 0.825420490228215226552899704968, 1.81016198886658620387128717269, 2.74213604250348583958439606487, 4.24593834905828010244639637140, 4.83606771764239126004594969628, 5.10082993873104227484754797963, 6.67217986610840586708393954408, 6.89544604540853082737536757711, 8.019966672938635264549316546568, 8.496848597033997215678695993508

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