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Properties

Label 2-1380-115.22-c1-0-5
Degree $2$
Conductor $1380$
Sign $0.0417 - 0.999i$
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 + (0.951 − 2.02i)5-s + (−3.48 + 3.48i)7-s − 1.00i·9-s + 2.73i·11-s + (−0.367 + 0.367i)13-s + (−0.757 − 2.10i)15-s + (−5.02 + 5.02i)17-s + 5.91·19-s + 4.92i·21-s + (−3.87 + 2.82i)23-s + (−3.18 − 3.85i)25-s + (−0.707 − 0.707i)27-s + 2.00i·29-s + 2.49·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.425 − 0.904i)5-s + (−1.31 + 1.31i)7-s − 0.333i·9-s + 0.823i·11-s + (−0.102 + 0.102i)13-s + (−0.195 − 0.543i)15-s + (−1.21 + 1.21i)17-s + 1.35·19-s + 1.07i·21-s + (−0.808 + 0.588i)23-s + (−0.637 − 0.770i)25-s + (−0.136 − 0.136i)27-s + 0.371i·29-s + 0.448·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0417 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0417 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0417 - 0.999i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.0417 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.115978835\)
\(L(\frac12)\) \(\approx\) \(1.115978835\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 + 2.02i)T \)
23 \( 1 + (3.87 - 2.82i)T \)
good7 \( 1 + (3.48 - 3.48i)T - 7iT^{2} \)
11 \( 1 - 2.73iT - 11T^{2} \)
13 \( 1 + (0.367 - 0.367i)T - 13iT^{2} \)
17 \( 1 + (5.02 - 5.02i)T - 17iT^{2} \)
19 \( 1 - 5.91T + 19T^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 - 2.49T + 31T^{2} \)
37 \( 1 + (6.38 - 6.38i)T - 37iT^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 + (-2.07 - 2.07i)T + 43iT^{2} \)
47 \( 1 + (-4.01 - 4.01i)T + 47iT^{2} \)
53 \( 1 + (-0.794 - 0.794i)T + 53iT^{2} \)
59 \( 1 - 5.00iT - 59T^{2} \)
61 \( 1 - 9.72iT - 61T^{2} \)
67 \( 1 + (2.09 - 2.09i)T - 67iT^{2} \)
71 \( 1 - 9.35T + 71T^{2} \)
73 \( 1 + (-8.18 + 8.18i)T - 73iT^{2} \)
79 \( 1 - 2.90T + 79T^{2} \)
83 \( 1 + (7.48 + 7.48i)T + 83iT^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (0.793 - 0.793i)T - 97iT^{2} \)
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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

−9.684453551125665979389261956264, −8.898106452592228586953322524261, −8.479880446192856480854293732040, −7.31642062618893778218681226210, −6.37586123377951764109017891249, −5.78705594315786366690250345368, −4.81052058238903323967976901350, −3.60696646421023591229970686302, −2.50735886724307470827487731090, −1.61745399040178821194472008964, 0.40797703685461012918654823070, 2.42967547108447068813460999629, 3.31707459644527157745608109015, 3.89851541195729030814040929847, 5.19037120353993393145351288824, 6.30315943077109799401289556253, 6.93840283303682343678576380866, 7.55702222556525627084569357996, 8.744695642009326960751523461149, 9.697854178098446090554549583325

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