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Properties

Label 2-1380-115.22-c1-0-4
Degree $2$
Conductor $1380$
Sign $-0.809 - 0.587i$
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 + (−0.577 + 2.16i)5-s + (−0.575 + 0.575i)7-s − 1.00i·9-s + 2.55i·11-s + (−1.98 + 1.98i)13-s + (1.11 + 1.93i)15-s + (−1.42 + 1.42i)17-s − 3.58·19-s + 0.813i·21-s + (0.644 − 4.75i)23-s + (−4.33 − 2.49i)25-s + (−0.707 − 0.707i)27-s − 3.90i·29-s − 7.98·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.258 + 0.966i)5-s + (−0.217 + 0.217i)7-s − 0.333i·9-s + 0.771i·11-s + (−0.549 + 0.549i)13-s + (0.289 + 0.499i)15-s + (−0.344 + 0.344i)17-s − 0.823·19-s + 0.177i·21-s + (0.134 − 0.990i)23-s + (−0.866 − 0.498i)25-s + (−0.136 − 0.136i)27-s − 0.725i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.809 - 0.587i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.809 - 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7502351396\)
\(L(\frac12)\) \(\approx\) \(0.7502351396\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.577 - 2.16i)T \)
23 \( 1 + (-0.644 + 4.75i)T \)
good7 \( 1 + (0.575 - 0.575i)T - 7iT^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
13 \( 1 + (1.98 - 1.98i)T - 13iT^{2} \)
17 \( 1 + (1.42 - 1.42i)T - 17iT^{2} \)
19 \( 1 + 3.58T + 19T^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + 7.98T + 31T^{2} \)
37 \( 1 + (5.07 - 5.07i)T - 37iT^{2} \)
41 \( 1 + 4.13T + 41T^{2} \)
43 \( 1 + (-5.16 - 5.16i)T + 43iT^{2} \)
47 \( 1 + (0.115 + 0.115i)T + 47iT^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 - 4.87iT - 59T^{2} \)
61 \( 1 - 5.54iT - 61T^{2} \)
67 \( 1 + (-5.83 + 5.83i)T - 67iT^{2} \)
71 \( 1 - 0.976T + 71T^{2} \)
73 \( 1 + (0.307 - 0.307i)T - 73iT^{2} \)
79 \( 1 + 6.52T + 79T^{2} \)
83 \( 1 + (-4.68 - 4.68i)T + 83iT^{2} \)
89 \( 1 + 3.06T + 89T^{2} \)
97 \( 1 + (5.38 - 5.38i)T - 97iT^{2} \)
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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

−9.891291106012851659460428300989, −9.086841112528654750066407756159, −8.221807598794432931878020615366, −7.36353011025419921520152938598, −6.75147160451133373265654745454, −6.08724653424678541111429922648, −4.68847917687529453804652093650, −3.81993960933419763772359600139, −2.67255844010039310721971805424, −1.94837472042817883220769722343, 0.26770637044222489935722331317, 1.88464069689602280800777824580, 3.28362134517895202433612789819, 4.00028061482123712605040392342, 5.08994057182850803495217938726, 5.64265197482246205974864541436, 6.98725271094011842543630936746, 7.75469511055562066043923185743, 8.661988625011749188481309248428, 9.078850847269915674525558871654

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