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Properties

Label 2-1380-92.91-c1-0-26
Degree $2$
Conductor $1380$
Sign $-0.210 - 0.977i$
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  + (−0.952 + 1.04i)2-s + i·3-s + (−0.186 − 1.99i)4-s − i·5-s + (−1.04 − 0.952i)6-s − 3.49·7-s + (2.25 + 1.70i)8-s − 9-s + (1.04 + 0.952i)10-s − 1.36·11-s + (1.99 − 0.186i)12-s + 3.35·13-s + (3.33 − 3.65i)14-s + 15-s + (−3.93 + 0.741i)16-s + 0.0719i·17-s + ⋯
L(s)  = 1  + (−0.673 + 0.739i)2-s + 0.577i·3-s + (−0.0930 − 0.995i)4-s − 0.447i·5-s + (−0.426 − 0.388i)6-s − 1.32·7-s + (0.798 + 0.601i)8-s − 0.333·9-s + (0.330 + 0.301i)10-s − 0.411·11-s + (0.574 − 0.0537i)12-s + 0.929·13-s + (0.890 − 0.977i)14-s + 0.258·15-s + (−0.982 + 0.185i)16-s + 0.0174i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.210 - 0.977i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8254800627\)
\(L(\frac12)\) \(\approx\) \(0.8254800627\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.952 - 1.04i)T \)
3 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (4.76 - 0.568i)T \)
good7 \( 1 + 3.49T + 7T^{2} \)
11 \( 1 + 1.36T + 11T^{2} \)
13 \( 1 - 3.35T + 13T^{2} \)
17 \( 1 - 0.0719iT - 17T^{2} \)
19 \( 1 - 5.04T + 19T^{2} \)
29 \( 1 - 9.80T + 29T^{2} \)
31 \( 1 + 8.07iT - 31T^{2} \)
37 \( 1 - 3.32iT - 37T^{2} \)
41 \( 1 - 1.37T + 41T^{2} \)
43 \( 1 + 8.91T + 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 4.71iT - 59T^{2} \)
61 \( 1 - 6.38iT - 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 15.7iT - 71T^{2} \)
73 \( 1 - 5.24T + 73T^{2} \)
79 \( 1 + 4.20T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 - 4.72iT - 97T^{2} \)
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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

−9.893184203917491959982258314229, −9.020909758383885875634120219036, −8.293840936197339794960040002650, −7.51314682068037346818152999412, −6.34247639592363588670986400894, −5.95437875363903521562480189732, −4.93242081601251356111205188199, −3.91722381908260855171803317267, −2.74144245430916207706973430025, −0.952728623851800862315193169462, 0.55445124865137421566500073982, 1.99789877577650650452049189439, 3.17161869481537185146873262036, 3.56969170184117978130718266354, 5.17455899613219926068611191170, 6.54351257308793456151467167458, 6.77986597816343009218542695795, 7.960638748534817654928663217117, 8.517730327940603716124503595405, 9.524785037135577016843879102932

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