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Properties

Label 2-1380-92.91-c1-0-23
Degree $2$
Conductor $1380$
Sign $0.336 - 0.941i$
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  + (−1.17 − 0.788i)2-s + i·3-s + (0.755 + 1.85i)4-s − i·5-s + (0.788 − 1.17i)6-s + 0.567·7-s + (0.574 − 2.76i)8-s − 9-s + (−0.788 + 1.17i)10-s − 2.37·11-s + (−1.85 + 0.755i)12-s − 2.37·13-s + (−0.665 − 0.447i)14-s + 15-s + (−2.85 + 2.79i)16-s − 1.13i·17-s + ⋯
L(s)  = 1  + (−0.829 − 0.557i)2-s + 0.577i·3-s + (0.377 + 0.925i)4-s − 0.447i·5-s + (0.322 − 0.479i)6-s + 0.214·7-s + (0.203 − 0.979i)8-s − 0.333·9-s + (−0.249 + 0.371i)10-s − 0.715·11-s + (−0.534 + 0.217i)12-s − 0.658·13-s + (−0.177 − 0.119i)14-s + 0.258·15-s + (−0.714 + 0.699i)16-s − 0.276i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.336 - 0.941i$
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: \((2,\ 1380,\ (\ :1/2),\ 0.336 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7780140688\)
\(L(\frac12)\) \(\approx\) \(0.7780140688\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.788i)T \)
3 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (-3.57 - 3.19i)T \)
good7 \( 1 - 0.567T + 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 + 1.13iT - 17T^{2} \)
19 \( 1 - 1.96T + 19T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 8.93iT - 37T^{2} \)
41 \( 1 + 0.300T + 41T^{2} \)
43 \( 1 - 3.96T + 43T^{2} \)
47 \( 1 + 0.733iT - 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 - 5.02iT - 59T^{2} \)
61 \( 1 - 9.36iT - 61T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 - 5.30iT - 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 3.71T + 79T^{2} \)
83 \( 1 + 7.56T + 83T^{2} \)
89 \( 1 - 14.2iT - 89T^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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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

−9.898631010151901884812186734889, −8.965000057915305843214743211542, −8.385933111464551043122766493658, −7.54396060333509170981392252678, −6.72945590300445324208837645274, −5.29386903097388098543142724895, −4.68975430777668827123933454755, −3.43012001252953819577690861648, −2.60787395136852090182551602341, −1.19738690224768899149239990351, 0.45604301790718446912503316867, 1.99391039484028163264316103499, 2.85815405979951157766100531360, 4.55911020485198007413586446548, 5.57088046564101081905972232586, 6.28135908087273533442028094762, 7.25632789645842918360124004066, 7.66418898599189611368367352022, 8.455702222839288347193921221549, 9.371356402043818143793530930095

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