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Properties

Label 2-2340-39.11-c1-0-13
Degree $2$
Conductor $2340$
Sign $0.0420 + 0.999i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  + (−0.707 + 0.707i)5-s + (−0.185 − 0.691i)7-s + (0.169 − 0.634i)11-s + (−3.37 + 1.26i)13-s + (1.60 − 2.77i)17-s + (6.36 − 1.70i)19-s + (−0.746 − 1.29i)23-s − 1.00i·25-s + (−8.21 + 4.74i)29-s + (−0.174 − 0.174i)31-s + (0.620 + 0.358i)35-s + (−1.64 − 0.439i)37-s + (−3.13 − 0.839i)41-s + (3.72 + 2.15i)43-s + (−9.26 − 9.26i)47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + (−0.0700 − 0.261i)7-s + (0.0512 − 0.191i)11-s + (−0.936 + 0.349i)13-s + (0.389 − 0.673i)17-s + (1.45 − 0.391i)19-s + (−0.155 − 0.269i)23-s − 0.200i·25-s + (−1.52 + 0.880i)29-s + (−0.0312 − 0.0312i)31-s + (0.104 + 0.0605i)35-s + (−0.269 − 0.0722i)37-s + (−0.489 − 0.131i)41-s + (0.568 + 0.328i)43-s + (−1.35 − 1.35i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0420 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0420 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0420 + 0.999i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.0420 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.109739169\)
\(L(\frac12)\) \(\approx\) \(1.109739169\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.37 - 1.26i)T \)
good7 \( 1 + (0.185 + 0.691i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.169 + 0.634i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.60 + 2.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.36 + 1.70i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.746 + 1.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.21 - 4.74i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.174 + 0.174i)T + 31iT^{2} \)
37 \( 1 + (1.64 + 0.439i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.13 + 0.839i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.72 - 2.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.26 + 9.26i)T + 47iT^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 + (-5.97 + 1.59i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.74 + 13.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 + 11.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.04 - 7.63i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-8.34 + 8.34i)T - 73iT^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + (7.18 - 7.18i)T - 83iT^{2} \)
89 \( 1 + (-2.68 + 10.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.46 - 1.19i)T + (84.0 - 48.5i)T^{2} \)
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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

−8.848330858215778798619869366902, −7.895421643395597856706608871338, −7.18633946591213717887671138377, −6.77341076702414618748795578174, −5.45597084314611352274855063547, −4.97100787594499171504160057749, −3.76683462423570669090275895008, −3.09485365539213550848882769465, −1.92797564267166506467641836731, −0.40525767339197142099197438527, 1.20733118078381779759757362712, 2.45090902861901002606205399384, 3.49132678004297915088518012079, 4.32287586808017366049209527716, 5.40222762129859156278294692713, 5.79471721588741040052659553905, 7.08291526822722106274823489335, 7.64334479956863569659491125190, 8.298950182426485228226218712499, 9.306173220404467221255721316292

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