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Properties

Label 2-1380-15.2-c1-0-15
Degree $2$
Conductor $1380$
Sign $-0.749 - 0.662i$
Analytic cond. $11.0193$
Root an. cond. $3.31954$
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  + (−0.292 + 1.70i)3-s + (2.12 + 0.707i)5-s + (3 + 3i)7-s + (−2.82 − i)9-s + 2.82i·11-s + (−3 + 3i)13-s + (−1.82 + 3.41i)15-s + (−1.41 + 1.41i)17-s − 4i·19-s + (−5.99 + 4.24i)21-s + (−0.707 − 0.707i)23-s + (3.99 + 3i)25-s + (2.53 − 4.53i)27-s − 2.82·29-s − 4·31-s + ⋯
L(s)  = 1  + (−0.169 + 0.985i)3-s + (0.948 + 0.316i)5-s + (1.13 + 1.13i)7-s + (−0.942 − 0.333i)9-s + 0.852i·11-s + (−0.832 + 0.832i)13-s + (−0.472 + 0.881i)15-s + (−0.342 + 0.342i)17-s − 0.917i·19-s + (−1.30 + 0.925i)21-s + (−0.147 − 0.147i)23-s + (0.799 + 0.600i)25-s + (0.487 − 0.872i)27-s − 0.525·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.810142656\)
\(L(\frac12)\) \(\approx\) \(1.810142656\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 - 1.70i)T \)
5 \( 1 + (-2.12 - 0.707i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-4 - 4i)T + 37iT^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-7 - 7i)T + 67iT^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (-12 - 12i)T + 97iT^{2} \)
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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

−9.780348964339415610426601576858, −9.151631210707736979347357479090, −8.674113387341247434807549996713, −7.42373515346045733079932940715, −6.47019542929063949458821509815, −5.46217985153158451086782595911, −4.99802435789963852958869090990, −4.13476556221293429666347153850, −2.54535488156820311773051452545, −2.00357536431767886759296281776, 0.74571688841136851292583895695, 1.69486811636955110913634629832, 2.78658373639082937498506049695, 4.24297701650165261364834819024, 5.35626394802433800821904376365, 5.81927688663643617064548646352, 6.92698186394244198211899544682, 7.72226299074687598536225802661, 8.187297904914736724391513683589, 9.210362155569832510305869059300

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