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Properties

Label 2-930-155.64-c1-0-8
Degree $2$
Conductor $930$
Sign $-0.179 - 0.983i$
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  + (−0.587 + 0.809i)2-s + (0.587 + 0.809i)3-s + (−0.309 − 0.951i)4-s + (0.898 − 2.04i)5-s − 6-s + (−2.04 + 0.664i)7-s + (0.951 + 0.309i)8-s + (−0.309 + 0.951i)9-s + (1.12 + 1.93i)10-s + (1.16 + 3.58i)11-s + (0.587 − 0.809i)12-s + (1.53 + 2.11i)13-s + (0.664 − 2.04i)14-s + (2.18 − 0.476i)15-s + (−0.809 + 0.587i)16-s + (−5.98 − 1.94i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (0.339 + 0.467i)3-s + (−0.154 − 0.475i)4-s + (0.401 − 0.915i)5-s − 0.408·6-s + (−0.772 + 0.250i)7-s + (0.336 + 0.109i)8-s + (−0.103 + 0.317i)9-s + (0.356 + 0.610i)10-s + (0.350 + 1.08i)11-s + (0.169 − 0.233i)12-s + (0.426 + 0.587i)13-s + (0.177 − 0.546i)14-s + (0.564 − 0.123i)15-s + (−0.202 + 0.146i)16-s + (−1.45 − 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.179 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821522 + 0.984687i\)
\(L(\frac12)\) \(\approx\) \(0.821522 + 0.984687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (-0.898 + 2.04i)T \)
31 \( 1 + (-5.56 - 0.110i)T \)
good7 \( 1 + (2.04 - 0.664i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.16 - 3.58i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.53 - 2.11i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.98 + 1.94i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.19 - 1.59i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.25 - 1.38i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.11 - 4.44i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 5.91iT - 37T^{2} \)
41 \( 1 + (-5.93 - 4.30i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.81 - 6.62i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-6.25 - 8.61i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.02 - 0.656i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.25 + 3.82i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 8.58iT - 67T^{2} \)
71 \( 1 + (-3.11 + 9.60i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.73 - 0.563i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.417 + 1.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.19 - 4.40i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.680 + 2.09i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.25 - 2.68i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.815928712325357469538789076746, −9.375530836097359405745985047422, −8.876531690093026413316049671068, −7.949908211391824135669712321651, −6.77898310419608877332433334186, −6.20472843604460381063572415029, −4.85814989595892062875493011849, −4.44598847002061985172326038588, −2.84357073478635095480504755674, −1.41782794453619807940539792820, 0.71629166500033261985258064799, 2.37765284992743175858749536402, 3.11140668662549643038621351067, 4.02289419759999630235141709575, 5.76745423873511298998306605457, 6.59775467835537857323939317929, 7.19358633639544871529249782196, 8.436026949136588669500964761684, 8.933009285838524443752895063214, 9.944081087097113383896281631705

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