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Properties

Label 2-1470-35.27-c1-0-38
Degree $2$
Conductor $1470$
Sign $-0.709 - 0.705i$
Analytic cond. $11.7380$
Root an. cond. $3.42607$
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.03 − 1.98i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−2.13 + 0.674i)10-s + 2.62·11-s + (0.707 − 0.707i)12-s + (−1.21 − 1.21i)13-s + (−2.13 + 0.674i)15-s − 1.00·16-s + (−5.35 + 5.35i)17-s + (0.707 − 0.707i)18-s − 4.65·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.460 − 0.887i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.674 + 0.213i)10-s + 0.791·11-s + (0.204 − 0.204i)12-s + (−0.337 − 0.337i)13-s + (−0.550 + 0.174i)15-s − 0.250·16-s + (−1.29 + 1.29i)17-s + (0.166 − 0.166i)18-s − 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.709 - 0.705i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.709 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2888232834\)
\(L(\frac12)\) \(\approx\) \(0.2888232834\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.03 + 1.98i)T \)
7 \( 1 \)
good11 \( 1 - 2.62T + 11T^{2} \)
13 \( 1 + (1.21 + 1.21i)T + 13iT^{2} \)
17 \( 1 + (5.35 - 5.35i)T - 17iT^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 + (3.62 - 3.62i)T - 23iT^{2} \)
29 \( 1 + 5.99iT - 29T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + (-2.79 - 2.79i)T + 37iT^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 + (0.545 - 0.545i)T - 43iT^{2} \)
47 \( 1 + (4.48 - 4.48i)T - 47iT^{2} \)
53 \( 1 + (6.06 - 6.06i)T - 53iT^{2} \)
59 \( 1 + 7.72T + 59T^{2} \)
61 \( 1 + 4.80iT - 61T^{2} \)
67 \( 1 + (1.81 + 1.81i)T + 67iT^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + (9.65 + 9.65i)T + 73iT^{2} \)
79 \( 1 - 8.99iT - 79T^{2} \)
83 \( 1 + (-7.99 - 7.99i)T + 83iT^{2} \)
89 \( 1 + 0.162T + 89T^{2} \)
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

−9.097720254097862581853185983130, −8.269795016547434661239745566524, −7.66847690518870597341878002774, −6.28131111488082471244452612801, −6.03886059571157290137329744522, −4.58076577265351255337000042391, −3.99804064175626342220161682877, −2.28486432952143093037849227018, −1.57402538541358185236009706214, −0.13681191912890855323913143255, 1.79315314826399952038874387605, 2.97257164644593458547715861995, 4.31343672828791016808715355439, 5.07458247685174320980747222458, 6.30276901008902829355513397528, 6.63739146454837496693903664339, 7.33750202938350315067619472279, 8.670283728686185396268194543324, 9.145103647656930763854458050033, 10.00294485087170000086435106511

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