Properties

Label 2-1380-69.68-c1-0-19
Degree $2$
Conductor $1380$
Sign $0.438 + 0.898i$
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  + (0.208 − 1.71i)3-s − 5-s + 0.187i·7-s + (−2.91 − 0.716i)9-s + 4.36·11-s + 6.75·13-s + (−0.208 + 1.71i)15-s − 0.956·17-s + 5.93i·19-s + (0.322 + 0.0390i)21-s + (4.02 − 2.60i)23-s + 25-s + (−1.83 + 4.85i)27-s − 4.44i·29-s − 3.97·31-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)3-s − 0.447·5-s + 0.0708i·7-s + (−0.971 − 0.238i)9-s + 1.31·11-s + 1.87·13-s + (−0.0538 + 0.443i)15-s − 0.231·17-s + 1.36i·19-s + (0.0703 + 0.00852i)21-s + (0.839 − 0.542i)23-s + 0.200·25-s + (−0.353 + 0.935i)27-s − 0.825i·29-s − 0.714·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.438 + 0.898i$
Analytic conductor: \(11.0193\)
Root analytic conductor: \(3.31954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :1/2),\ 0.438 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.793987407\)
\(L(\frac12)\) \(\approx\) \(1.793987407\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.208 + 1.71i)T \)
5 \( 1 + T \)
23 \( 1 + (-4.02 + 2.60i)T \)
good7 \( 1 - 0.187iT - 7T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 - 6.75T + 13T^{2} \)
17 \( 1 + 0.956T + 17T^{2} \)
19 \( 1 - 5.93iT - 19T^{2} \)
29 \( 1 + 4.44iT - 29T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 - 9.51iT - 37T^{2} \)
41 \( 1 + 8.47iT - 41T^{2} \)
43 \( 1 + 7.63iT - 43T^{2} \)
47 \( 1 + 7.92iT - 47T^{2} \)
53 \( 1 - 6.24T + 53T^{2} \)
59 \( 1 - 0.525iT - 59T^{2} \)
61 \( 1 + 2.02iT - 61T^{2} \)
67 \( 1 + 7.87iT - 67T^{2} \)
71 \( 1 - 2.38iT - 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 - 4.08T + 83T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−8.980600524987397587946936398265, −8.693041059111897807300719885100, −7.87077927743011231065610674930, −6.88627221866378549263359338003, −6.32556316816697862465993761411, −5.53581384628983151979471973022, −3.98382609756473828072479668656, −3.44412855357633381714337068276, −1.92175281528630338626049885850, −0.935513714152476071302327916409, 1.16711068157779886149143845328, 2.94599103915216169221146162287, 3.80878266317208377213252974588, 4.39196271000826870899345662127, 5.50552430725582737835401209564, 6.38915952280230522850644880314, 7.25892979011162994875019266869, 8.451811763120717701207030350843, 9.018116434217975278019657718781, 9.438952601715242957277832845676

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