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Properties

Label 2-1380-115.68-c1-0-1
Degree $2$
Conductor $1380$
Sign $-0.956 - 0.290i$
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 + (−1.66 − 4.49i)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 + (−0.346 − 0.938i)23-s + (0.187 − 0.982i)25-s + (−0.136 + 0.136i)27-s + 1.34i·29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.956 - 0.290i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7303789314\)
\(L(\frac12)\) \(\approx\) \(0.7303789314\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 4.49i)T \)
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

−10.07378593912379650563808877976, −8.973637957679722203742993691096, −8.487993861354035879562968117920, −7.48884032457893470725891371570, −6.71347871386746757231972304796, −6.06415530819378780357916145292, −4.52878793013511933013230750048, −3.80978277658372693808997189663, −3.27229423927308997910247733243, −1.77541307591170347316616549180, 0.27514993829756336152891137787, 1.76519646242132686845518973652, 3.22598772433079559973147742745, 3.72221759518922313439987982996, 5.10902572422429328220593339360, 5.88752606379681150593210597924, 6.79978638450873668207422555268, 7.895230775415683217743406016922, 8.280910494714922075775086907089, 9.068164466631059019140543974408

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