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Properties

Label 2-1380-5.4-c1-0-16
Degree $2$
Conductor $1380$
Sign $-0.354 + 0.935i$
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  + i·3-s + (0.792 − 2.09i)5-s − 4.31i·7-s − 9-s − 2.26·11-s + 2.25i·13-s + (2.09 + 0.792i)15-s − 0.535i·17-s + 6.91·19-s + 4.31·21-s − i·23-s + (−3.74 − 3.31i)25-s − i·27-s − 5.64·29-s − 10.4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.354 − 0.935i)5-s − 1.63i·7-s − 0.333·9-s − 0.683·11-s + 0.625i·13-s + (0.539 + 0.204i)15-s − 0.129i·17-s + 1.58·19-s + 0.941·21-s − 0.208i·23-s + (−0.748 − 0.663i)25-s − 0.192i·27-s − 1.04·29-s − 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.354 + 0.935i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.354 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248485641\)
\(L(\frac12)\) \(\approx\) \(1.248485641\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-0.792 + 2.09i)T \)
23 \( 1 + iT \)
good7 \( 1 + 4.31iT - 7T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 + 0.535iT - 17T^{2} \)
19 \( 1 - 6.91T + 19T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 8.10iT - 37T^{2} \)
41 \( 1 - 0.633T + 41T^{2} \)
43 \( 1 + 1.30iT - 43T^{2} \)
47 \( 1 + 12.4iT - 47T^{2} \)
53 \( 1 - 5.98iT - 53T^{2} \)
59 \( 1 + 7.01T + 59T^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 - 0.151T + 71T^{2} \)
73 \( 1 - 8.88iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 0.716iT - 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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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

−9.467983172153934001123145342113, −8.692151136661730213214138270044, −7.55930220863463463379072579015, −7.17501017521962754720197085082, −5.72795200241704564231925182478, −5.10187145586676224286571032011, −4.17463277011319655216424176302, −3.48987830904033928463517670128, −1.84469953257396575843046629761, −0.48994670740188908062277370915, 1.74999887420823075269056803524, 2.72696382054925383135967301274, 3.34350324342048817100290247115, 5.30136580111789050052204531002, 5.60250583405215266672215017413, 6.48303410806807807723773136500, 7.51688659277215841350566075566, 8.005847112986606345996352695093, 9.166802405043603876988236430483, 9.637950239161227920802145183454

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