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Properties

Label 2-1380-92.91-c1-0-8
Degree $2$
Conductor $1380$
Sign $-0.299 + 0.954i$
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  + (0.800 + 1.16i)2-s + i·3-s + (−0.719 + 1.86i)4-s − i·5-s + (−1.16 + 0.800i)6-s + 1.17·7-s + (−2.75 + 0.653i)8-s − 9-s + (1.16 − 0.800i)10-s − 4.08·11-s + (−1.86 − 0.719i)12-s − 5.99·13-s + (0.942 + 1.37i)14-s + 15-s + (−2.96 − 2.68i)16-s − 0.761i·17-s + ⋯
L(s)  = 1  + (0.565 + 0.824i)2-s + 0.577i·3-s + (−0.359 + 0.932i)4-s − 0.447i·5-s + (−0.476 + 0.326i)6-s + 0.445·7-s + (−0.972 + 0.231i)8-s − 0.333·9-s + (0.368 − 0.253i)10-s − 1.23·11-s + (−0.538 − 0.207i)12-s − 1.66·13-s + (0.252 + 0.367i)14-s + 0.258·15-s + (−0.740 − 0.671i)16-s − 0.184i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.299 + 0.954i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3414129874\)
\(L(\frac12)\) \(\approx\) \(0.3414129874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.800 - 1.16i)T \)
3 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (4.78 - 0.308i)T \)
good7 \( 1 - 1.17T + 7T^{2} \)
11 \( 1 + 4.08T + 11T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 + 0.761iT - 17T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 + 1.24iT - 37T^{2} \)
41 \( 1 + 1.89T + 41T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 - 2.61iT - 47T^{2} \)
53 \( 1 + 2.04iT - 53T^{2} \)
59 \( 1 - 1.92iT - 59T^{2} \)
61 \( 1 - 4.50iT - 61T^{2} \)
67 \( 1 + 5.89T + 67T^{2} \)
71 \( 1 - 9.16iT - 71T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 - 7.39T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 + 5.05iT - 97T^{2} \)
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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

−9.787710213234679059740102932839, −9.449410350852548731058693523188, −8.221178845146601002027726794155, −7.75798309257350969812792519700, −7.01913059615712283830139054957, −5.59612364087477166702064092504, −5.20664824981870483272198220470, −4.54635682529171574810514963714, −3.40986655018036733498332616852, −2.36006399382492870794957136776, 0.10397060468456878308249076496, 1.84806619561498333065375186319, 2.60606034146190412803268864803, 3.55968626499914185602206935687, 4.94800173256546552271842236158, 5.35241434312684780125600745625, 6.43725999655178240610538635488, 7.52988572801840136840772175280, 7.952466348215505516480945522851, 9.340960945046517699268740374434

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