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Properties

Label 2-930-155.129-c1-0-28
Degree $2$
Conductor $930$
Sign $0.991 + 0.133i$
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 + (0.866 + 0.5i)3-s − 4-s + (−0.133 − 2.23i)5-s + (−0.5 + 0.866i)6-s + (2.29 + 1.32i)7-s − i·8-s + (0.499 + 0.866i)9-s + (2.23 − 0.133i)10-s + (−2.32 − 4.02i)11-s + (−0.866 − 0.5i)12-s + (4.58 − 2.64i)13-s + (−1.32 + 2.29i)14-s + (1 − 1.99i)15-s + 16-s + (−5.19 − 3i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (−0.0599 − 0.998i)5-s + (−0.204 + 0.353i)6-s + (0.866 + 0.499i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.705 − 0.0423i)10-s + (−0.700 − 1.21i)11-s + (−0.249 − 0.144i)12-s + (1.27 − 0.733i)13-s + (−0.353 + 0.612i)14-s + (0.258 − 0.516i)15-s + 0.250·16-s + (−1.26 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.991 + 0.133i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.991 + 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81794 - 0.121902i\)
\(L(\frac12)\) \(\approx\) \(1.81794 - 0.121902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
31 \( 1 + (4.14 - 3.71i)T \)
good7 \( 1 + (-2.29 - 1.32i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.32 + 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.58 + 2.64i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.64 + 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.29iT - 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.64 - 6.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 + 0.354i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.29iT - 47T^{2} \)
53 \( 1 + (-7.18 + 4.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.32 + 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (-0.613 + 0.354i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.29 - 12.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12.6 + 7.29i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.29 - 5.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.0 - 5.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.29T + 89T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−9.733156125124835262665187337665, −8.726886487839887313173119565956, −8.543258713294646961798453822512, −7.87875302595902888795178870006, −6.61193006745750085083811750155, −5.39181661124983247993247585583, −5.07080223969237564466757499310, −3.91272133387558837453531346680, −2.64656862750657209978385016557, −0.875964872175594955536393592779, 1.63423821052126488316390052990, 2.36972941425763651711999232065, 3.78168403713450981818791327351, 4.30735935376867027161971768853, 5.76484561168682224066290903108, 6.88190469443419717248890252984, 7.64450781653681412817991489829, 8.367756174956679589155302776000, 9.379189947888657496943086089157, 10.25956505560947064099035918985

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