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Properties

Label 2-1380-115.22-c1-0-2
Degree $2$
Conductor $1380$
Sign $-0.933 - 0.358i$
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 + (−1.62 + 1.53i)5-s + (2.42 − 2.42i)7-s − 1.00i·9-s − 4.29i·11-s + (−2.97 + 2.97i)13-s + (0.0580 − 2.23i)15-s + (−5.55 + 5.55i)17-s + 3.90·19-s + 3.42i·21-s + (3.00 + 3.74i)23-s + (0.259 − 4.99i)25-s + (0.707 + 0.707i)27-s + 6.44i·29-s − 7.44·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.725 + 0.688i)5-s + (0.915 − 0.915i)7-s − 0.333i·9-s − 1.29i·11-s + (−0.825 + 0.825i)13-s + (0.0149 − 0.577i)15-s + (−1.34 + 1.34i)17-s + 0.896·19-s + 0.747i·21-s + (0.625 + 0.780i)23-s + (0.0519 − 0.998i)25-s + (0.136 + 0.136i)27-s + 1.19i·29-s − 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.933 - 0.358i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4740909348\)
\(L(\frac12)\) \(\approx\) \(0.4740909348\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 1.53i)T \)
23 \( 1 + (-3.00 - 3.74i)T \)
good7 \( 1 + (-2.42 + 2.42i)T - 7iT^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
13 \( 1 + (2.97 - 2.97i)T - 13iT^{2} \)
17 \( 1 + (5.55 - 5.55i)T - 17iT^{2} \)
19 \( 1 - 3.90T + 19T^{2} \)
29 \( 1 - 6.44iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + (1.45 - 1.45i)T - 37iT^{2} \)
41 \( 1 + 4.04T + 41T^{2} \)
43 \( 1 + (-3.37 - 3.37i)T + 43iT^{2} \)
47 \( 1 + (7.41 + 7.41i)T + 47iT^{2} \)
53 \( 1 + (-0.484 - 0.484i)T + 53iT^{2} \)
59 \( 1 - 15.1iT - 59T^{2} \)
61 \( 1 + 12.1iT - 61T^{2} \)
67 \( 1 + (1.35 - 1.35i)T - 67iT^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + (8.24 - 8.24i)T - 73iT^{2} \)
79 \( 1 + 7.52T + 79T^{2} \)
83 \( 1 + (3.22 + 3.22i)T + 83iT^{2} \)
89 \( 1 + 7.86T + 89T^{2} \)
97 \( 1 + (4.32 - 4.32i)T - 97iT^{2} \)
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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

−10.21047778056746937045925520715, −9.025592964186472427974379160117, −8.357853225263695681781850552499, −7.32382594972060253018793565934, −6.89154919030386128067385312648, −5.75819027843860723840110453413, −4.74613999841933118220811643511, −3.98248043884845191193533134413, −3.18369747597523637791954450374, −1.52621116312352065390037878335, 0.20346661359070545217068988689, 1.79969604998265937078615796002, 2.78114430928202061831735506617, 4.54017097663113108442312855361, 4.88657864081187155624152474801, 5.66353647198074032327234813451, 7.13167768241186248404181683450, 7.44472044142042263884951470233, 8.385958311595164132158084535374, 9.135347081653984129353824617681

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