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Properties

Label 2-1380-115.68-c1-0-13
Degree $2$
Conductor $1380$
Sign $0.842 - 0.538i$
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  + (0.707 + 0.707i)3-s + (1.72 − 1.42i)5-s + (1.66 + 1.66i)7-s + 1.00i·9-s + 0.0251i·11-s + (4.31 + 4.31i)13-s + (2.22 + 0.210i)15-s + (−2.93 − 2.93i)17-s + 5.73·19-s + 2.35i·21-s + (−4.49 − 1.66i)23-s + (0.938 − 4.91i)25-s + (−0.707 + 0.707i)27-s + 7.21i·29-s − 8.64·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.770 − 0.637i)5-s + (0.630 + 0.630i)7-s + 0.333i·9-s + 0.00757i·11-s + (1.19 + 1.19i)13-s + (0.574 + 0.0544i)15-s + (−0.711 − 0.711i)17-s + 1.31·19-s + 0.514i·21-s + (−0.938 − 0.346i)23-s + (0.187 − 0.982i)25-s + (−0.136 + 0.136i)27-s + 1.34i·29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.489229081\)
\(L(\frac12)\) \(\approx\) \(2.489229081\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.72 + 1.42i)T \)
23 \( 1 + (4.49 + 1.66i)T \)
good7 \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \)
11 \( 1 - 0.0251iT - 11T^{2} \)
13 \( 1 + (-4.31 - 4.31i)T + 13iT^{2} \)
17 \( 1 + (2.93 + 2.93i)T + 17iT^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
29 \( 1 - 7.21iT - 29T^{2} \)
31 \( 1 + 8.64T + 31T^{2} \)
37 \( 1 + (-6.47 - 6.47i)T + 37iT^{2} \)
41 \( 1 + 0.0620T + 41T^{2} \)
43 \( 1 + (-5.43 + 5.43i)T - 43iT^{2} \)
47 \( 1 + (4.36 - 4.36i)T - 47iT^{2} \)
53 \( 1 + (-9.27 + 9.27i)T - 53iT^{2} \)
59 \( 1 - 2.10iT - 59T^{2} \)
61 \( 1 + 4.81iT - 61T^{2} \)
67 \( 1 + (0.147 + 0.147i)T + 67iT^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 + (1.58 + 1.58i)T + 73iT^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 + (0.182 - 0.182i)T - 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-12.8 - 12.8i)T + 97iT^{2} \)
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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

−9.386924595671451946133648952371, −8.940441701482973409823348718994, −8.388073277002800021145816775112, −7.25955208286519874108182717536, −6.23016514234166189859709978568, −5.36444264007773985038922182290, −4.67764320763819581991312577786, −3.65068310292515267246080424760, −2.32511471155904374469172852282, −1.45070791609846784652730985344, 1.13649043297460322128786430104, 2.20525579888258194686521158055, 3.33457618309566676372983597708, 4.19310045964558037249647005462, 5.74917275670992924308612497246, 5.99131060737764730940947480516, 7.31830593051754411877529878094, 7.72388606529589031112997853664, 8.642141263370761672971836790475, 9.536028320625579394649756098179

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