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Properties

Label 2-930-93.68-c1-0-32
Degree $2$
Conductor $930$
Sign $0.970 + 0.239i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + i·2-s + (1.28 − 1.15i)3-s − 4-s + (−0.866 − 0.5i)5-s + (1.15 + 1.28i)6-s + (0.615 + 1.06i)7-s − i·8-s + (0.318 − 2.98i)9-s + (0.5 − 0.866i)10-s + (0.427 − 0.741i)11-s + (−1.28 + 1.15i)12-s + (4.88 + 2.82i)13-s + (−1.06 + 0.615i)14-s + (−1.69 + 0.358i)15-s + 16-s + (−3.42 − 5.93i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.743 − 0.668i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.472 + 0.525i)6-s + (0.232 + 0.402i)7-s − 0.353i·8-s + (0.106 − 0.994i)9-s + (0.158 − 0.273i)10-s + (0.129 − 0.223i)11-s + (−0.371 + 0.334i)12-s + (1.35 + 0.782i)13-s + (−0.284 + 0.164i)14-s + (−0.437 + 0.0926i)15-s + 0.250·16-s + (−0.830 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.970 + 0.239i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.970 + 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90188 - 0.231097i\)
\(L(\frac12)\) \(\approx\) \(1.90188 - 0.231097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.28 + 1.15i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (1.91 + 5.22i)T \)
good7 \( 1 + (-0.615 - 1.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.427 + 0.741i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.88 - 2.82i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.42 + 5.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.242 + 0.420i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.95T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
37 \( 1 + (-7.56 + 4.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.21 - 3.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.37 - 2.52i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.88iT - 47T^{2} \)
53 \( 1 + (-0.346 + 0.599i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.639 + 0.369i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.48iT - 61T^{2} \)
67 \( 1 + (-4.07 + 7.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.36 - 1.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.44 + 2.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.99 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.20 - 2.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.45T + 89T^{2} \)
97 \( 1 + 5.55T + 97T^{2} \)
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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

−9.388922843297739265632020229042, −9.079202423000100528593784226628, −8.365502647413562110490969624706, −7.48259046093083627759319210740, −6.77514583884105618373087775871, −5.95243692509861429256937751275, −4.72524825429583053366074716388, −3.75478353547582810132163552065, −2.54023865702820616978014293707, −0.983121596782615690883047391264, 1.41763678135882855777240784782, 2.83532257927227078640596006184, 3.73070653054153331241413415649, 4.35402463736795353271545409110, 5.49860156771492110579988196436, 6.79427620471771634249091064268, 7.961088888473061181992825144157, 8.544303143981941229943699079982, 9.238206183423199931474767854320, 10.34418736711554340027607395800

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