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Properties

Label 2-1380-92.91-c1-0-37
Degree $2$
Conductor $1380$
Sign $-0.522 - 0.852i$
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  + (−0.160 + 1.40i)2-s + i·3-s + (−1.94 − 0.451i)4-s + i·5-s + (−1.40 − 0.160i)6-s − 1.44·7-s + (0.947 − 2.66i)8-s − 9-s + (−1.40 − 0.160i)10-s + 4.31·11-s + (0.451 − 1.94i)12-s + 5.03·13-s + (0.232 − 2.03i)14-s − 15-s + (3.59 + 1.75i)16-s − 3.32i·17-s + ⋯
L(s)  = 1  + (−0.113 + 0.993i)2-s + 0.577i·3-s + (−0.974 − 0.225i)4-s + 0.447i·5-s + (−0.573 − 0.0655i)6-s − 0.546·7-s + (0.334 − 0.942i)8-s − 0.333·9-s + (−0.444 − 0.0507i)10-s + 1.30·11-s + (0.130 − 0.562i)12-s + 1.39·13-s + (0.0620 − 0.543i)14-s − 0.258·15-s + (0.898 + 0.439i)16-s − 0.806i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544761408\)
\(L(\frac12)\) \(\approx\) \(1.544761408\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.160 - 1.40i)T \)
3 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + (-3.36 - 3.41i)T \)
good7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 - 5.03T + 13T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
29 \( 1 - 0.0416T + 29T^{2} \)
31 \( 1 - 4.91iT - 31T^{2} \)
37 \( 1 + 7.69iT - 37T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 1.31T + 43T^{2} \)
47 \( 1 + 3.27iT - 47T^{2} \)
53 \( 1 - 3.82iT - 53T^{2} \)
59 \( 1 - 9.30iT - 59T^{2} \)
61 \( 1 + 3.55iT - 61T^{2} \)
67 \( 1 - 8.24T + 67T^{2} \)
71 \( 1 - 2.13iT - 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 - 8.18T + 83T^{2} \)
89 \( 1 - 9.95iT - 89T^{2} \)
97 \( 1 - 0.397iT - 97T^{2} \)
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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

−9.429285414172159240856654575710, −9.234611435982375350194086177636, −8.302645462278566096360611007795, −7.18924888935784875815407516986, −6.67029756724173125048634274727, −5.79398437037341694109655565083, −5.01788916216350881346312427736, −3.74820372340474376605122635797, −3.34731048243552385628196001261, −1.12260328807372196850567592061, 0.905181931081330667051082895664, 1.68969947314560488114796536149, 3.20592592865015204452732488813, 3.80231167605780369060748150431, 4.95237500823762553581907281683, 6.04795595300949965694482875338, 6.75745596693698557820747073819, 8.045371695475764505593908867671, 8.615301959342386804516303931852, 9.359764613412830536069724806865

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