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Properties

Label 2-1380-92.91-c1-0-89
Degree $2$
Conductor $1380$
Sign $-0.965 + 0.260i$
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  + (1.16 − 0.804i)2-s + i·3-s + (0.707 − 1.87i)4-s + i·5-s + (0.804 + 1.16i)6-s − 4.03·7-s + (−0.681 − 2.74i)8-s − 9-s + (0.804 + 1.16i)10-s − 0.402·11-s + (1.87 + 0.707i)12-s + 0.442·13-s + (−4.69 + 3.24i)14-s − 15-s + (−2.99 − 2.64i)16-s − 4.35i·17-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)2-s + 0.577i·3-s + (0.353 − 0.935i)4-s + 0.447i·5-s + (0.328 + 0.474i)6-s − 1.52·7-s + (−0.240 − 0.970i)8-s − 0.333·9-s + (0.254 + 0.367i)10-s − 0.121·11-s + (0.540 + 0.204i)12-s + 0.122·13-s + (−1.25 + 0.867i)14-s − 0.258·15-s + (−0.749 − 0.661i)16-s − 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.965 + 0.260i$
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: \((2,\ 1380,\ (\ :1/2),\ -0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7692266492\)
\(L(\frac12)\) \(\approx\) \(0.7692266492\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.804i)T \)
3 \( 1 - iT \)
5 \( 1 - iT \)
23 \( 1 + (2.80 + 3.88i)T \)
good7 \( 1 + 4.03T + 7T^{2} \)
11 \( 1 + 0.402T + 11T^{2} \)
13 \( 1 - 0.442T + 13T^{2} \)
17 \( 1 + 4.35iT - 17T^{2} \)
19 \( 1 + 3.70T + 19T^{2} \)
29 \( 1 + 0.591T + 29T^{2} \)
31 \( 1 + 3.29iT - 31T^{2} \)
37 \( 1 + 0.0841iT - 37T^{2} \)
41 \( 1 + 9.40T + 41T^{2} \)
43 \( 1 + 0.681T + 43T^{2} \)
47 \( 1 + 5.37iT - 47T^{2} \)
53 \( 1 + 0.279iT - 53T^{2} \)
59 \( 1 - 12.4iT - 59T^{2} \)
61 \( 1 - 1.46iT - 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 - 9.25T + 73T^{2} \)
79 \( 1 - 0.0927T + 79T^{2} \)
83 \( 1 - 3.36T + 83T^{2} \)
89 \( 1 - 5.03iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−9.533081611838617608514856102542, −8.722772767343131160979520010485, −7.26316631946838931776957941779, −6.47157791422778984347184819823, −5.89712750944745886201614685273, −4.81249809125784072174058241648, −3.88047178442751959606182506599, −3.12967344584108757042037646086, −2.32478384589005704505105031189, −0.21369233878907933729068463453, 1.91667940847937481587213498701, 3.19486066979372680884173116942, 3.87594556133916645472066743275, 5.05900285873150240304942758107, 6.11266200848243227073052931744, 6.39809344399459378291308443267, 7.33861552610782205374994492849, 8.214717600589421525713204846240, 8.888233870480739289684854319867, 9.872460058979799903208423436748

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