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Properties

Label 2-1470-21.20-c1-0-34
Degree $2$
Conductor $1470$
Sign $-0.387 + 0.921i$
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  − i·2-s + (−0.767 − 1.55i)3-s − 4-s + 5-s + (−1.55 + 0.767i)6-s + i·8-s + (−1.82 + 2.38i)9-s − i·10-s + 2.30i·11-s + (0.767 + 1.55i)12-s + 5.00i·13-s + (−0.767 − 1.55i)15-s + 16-s + 6.30·17-s + (2.38 + 1.82i)18-s − 7.55i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.443 − 0.896i)3-s − 0.5·4-s + 0.447·5-s + (−0.633 + 0.313i)6-s + 0.353i·8-s + (−0.607 + 0.794i)9-s − 0.316i·10-s + 0.695i·11-s + (0.221 + 0.448i)12-s + 1.38i·13-s + (−0.198 − 0.400i)15-s + 0.250·16-s + 1.52·17-s + (0.561 + 0.429i)18-s − 1.73i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.387 + 0.921i$
Analytic conductor: \(11.7380\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1470} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ -0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424585824\)
\(L(\frac12)\) \(\approx\) \(1.424585824\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.767 + 1.55i)T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 - 2.30iT - 11T^{2} \)
13 \( 1 - 5.00iT - 13T^{2} \)
17 \( 1 - 6.30T + 17T^{2} \)
19 \( 1 + 7.55iT - 19T^{2} \)
23 \( 1 + 6.37iT - 23T^{2} \)
29 \( 1 + 3.83iT - 29T^{2} \)
31 \( 1 + 4.38iT - 31T^{2} \)
37 \( 1 + 6.90T + 37T^{2} \)
41 \( 1 - 9.79T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 + 3.25iT - 53T^{2} \)
59 \( 1 - 9.93T + 59T^{2} \)
61 \( 1 - 5.98iT - 61T^{2} \)
67 \( 1 + 2.76T + 67T^{2} \)
71 \( 1 - 2.85iT - 71T^{2} \)
73 \( 1 + 3.68iT - 73T^{2} \)
79 \( 1 - 2.55T + 79T^{2} \)
83 \( 1 + 1.83T + 83T^{2} \)
89 \( 1 - 5.88T + 89T^{2} \)
97 \( 1 + 4.61iT - 97T^{2} \)
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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

−9.326926226547560094969401565991, −8.595851982596770959294558218106, −7.51904653687339224709035453316, −6.85665526239358754828543851543, −5.99181560599461329204498989271, −5.02794736562303663231499745223, −4.22100272682396287940839321027, −2.68193185665452314127507843500, −1.99862925793882050797844617872, −0.76129457649956699634097071110, 1.08084130400922870159618226030, 3.23102287490192380751070775859, 3.70836700973831407220767881739, 5.19206937881523218033895364713, 5.59696225341348832058030625232, 6.11131506743872871509870857198, 7.43557097700145146982527779239, 8.137114745295588409708008710155, 8.964398707586336721065460532466, 9.871279326973749687022560961244

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