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Properties

Label 2-3060-15.8-c1-0-30
Degree $2$
Conductor $3060$
Sign $-0.944 + 0.329i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4227296108\)
\(L(\frac12)\) \(\approx\) \(0.4227296108\)
\(L(\frac{3}{2})\) 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

−8.140868546446710622788214266129, −7.74304733431856483281511407671, −6.98169661301708854721640144542, −6.06644054248241022280810463761, −5.45002507673881898893629528615, −4.26761142948732883254728279730, −3.55862077829921941387965758107, −2.86308839211629922473748734843, −1.48224433569450462248157669355, −0.12720317096430390668781971069, 1.64969760566444015348219130448, 2.20099946113478865337906243315, 3.70661325455982395445991490341, 4.58335044289715601832035968722, 4.97424093956184658143277075854, 5.89157993694767841212997901356, 6.92168745248821152505293335447, 7.75283432293717627516174390768, 8.166465482850085358011118237931, 9.060469519433748025904264549247

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