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Properties

Label 2-1380-92.91-c1-0-79
Degree $2$
Conductor $1380$
Sign $-0.979 + 0.199i$
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  + (−1.18 − 0.767i)2-s − i·3-s + (0.820 + 1.82i)4-s − i·5-s + (−0.767 + 1.18i)6-s + 0.597·7-s + (0.425 − 2.79i)8-s − 9-s + (−0.767 + 1.18i)10-s − 3.20·11-s + (1.82 − 0.820i)12-s + 3.38·13-s + (−0.709 − 0.458i)14-s − 15-s + (−2.65 + 2.99i)16-s − 7.02i·17-s + ⋯
L(s)  = 1  + (−0.839 − 0.542i)2-s − 0.577i·3-s + (0.410 + 0.911i)4-s − 0.447i·5-s + (−0.313 + 0.484i)6-s + 0.225·7-s + (0.150 − 0.988i)8-s − 0.333·9-s + (−0.242 + 0.375i)10-s − 0.966·11-s + (0.526 − 0.236i)12-s + 0.940·13-s + (−0.189 − 0.122i)14-s − 0.258·15-s + (−0.663 + 0.748i)16-s − 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.979 + 0.199i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ -0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7784820942\)
\(L(\frac12)\) \(\approx\) \(0.7784820942\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.767i)T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
23 \( 1 + (1.05 - 4.67i)T \)
good7 \( 1 - 0.597T + 7T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 + 7.02iT - 17T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
29 \( 1 + 7.11T + 29T^{2} \)
31 \( 1 + 7.60iT - 31T^{2} \)
37 \( 1 + 2.75iT - 37T^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
43 \( 1 - 1.14T + 43T^{2} \)
47 \( 1 - 5.36iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 0.724iT - 59T^{2} \)
61 \( 1 + 7.28iT - 61T^{2} \)
67 \( 1 + 2.69T + 67T^{2} \)
71 \( 1 + 2.14iT - 71T^{2} \)
73 \( 1 - 8.52T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 0.284T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 6.87iT - 97T^{2} \)
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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

−9.470201860738598506084567852952, −8.330878133959585582433046144817, −7.68212061820227303514689163199, −7.21115870896752423054845975726, −5.92277726985646872711717921282, −5.06312083133978694390233797354, −3.70098545256822337831141802338, −2.72469720226648769436196340093, −1.61955628895847952270182657510, −0.43198818025386255848925553106, 1.48747151446386557977135550293, 2.82695562481634895387092413588, 3.99354720308302480449141115261, 5.23331685332262184970002614249, 5.87897586633748208547575966554, 6.73285333417135199492670572227, 7.75994796833912962233446149328, 8.323649856736287411089529763981, 9.065116554016478990766290835737, 9.998869322018376951013377188031

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