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Properties

Label 2-930-93.68-c1-0-42
Degree $2$
Conductor $930$
Sign $-0.474 - 0.880i$
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 + (0.761 − 1.55i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−1.55 − 0.761i)6-s + (−0.777 − 1.34i)7-s + i·8-s + (−1.84 − 2.36i)9-s + (−0.5 + 0.866i)10-s + (−1.42 + 2.47i)11-s + (−0.761 + 1.55i)12-s + (−1.67 − 0.966i)13-s + (−1.34 + 0.777i)14-s + (−1.43 + 0.966i)15-s + 16-s + (−2.05 − 3.55i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.439 − 0.898i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (−0.635 − 0.310i)6-s + (−0.293 − 0.508i)7-s + 0.353i·8-s + (−0.613 − 0.789i)9-s + (−0.158 + 0.273i)10-s + (−0.430 + 0.745i)11-s + (−0.219 + 0.449i)12-s + (−0.464 − 0.268i)13-s + (−0.359 + 0.207i)14-s + (−0.371 + 0.249i)15-s + 0.250·16-s + (−0.497 − 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.474 - 0.880i$
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: \((2,\ 930,\ (\ :1/2),\ -0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299170 + 0.501000i\)
\(L(\frac12)\) \(\approx\) \(0.299170 + 0.501000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.761 + 1.55i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-4.24 + 3.60i)T \)
good7 \( 1 + (0.777 + 1.34i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.42 - 2.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.67 + 0.966i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.05 + 3.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.49 - 4.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.12T + 23T^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
37 \( 1 + (2.22 - 1.28i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.09 - 0.629i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.09 - 5.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 + (5.37 - 9.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.5 + 6.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.90iT - 61T^{2} \)
67 \( 1 + (-0.173 + 0.300i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0462 + 0.0266i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (14.2 + 8.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.174 - 0.100i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 2.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.22T + 89T^{2} \)
97 \( 1 - 0.201T + 97T^{2} \)
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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

−9.823822340585839680397164639026, −8.563167517545807346770487241019, −7.84734867600595797390763817529, −7.21173418625510092503747241640, −6.12254712586768617326556583691, −4.89255086802034190619609355663, −3.84864515623028278314824295522, −2.81906528504403129511820638829, −1.74589474120706574982580441269, −0.24760596838148024031882079392, 2.49614013853481846943034292867, 3.53317817173765740169774494390, 4.47909359871708238955497885980, 5.41092674190270289308021139099, 6.27606401177405448826969481602, 7.33390905581987596478143601329, 8.392726895781598610151971377133, 8.672309883959370436652297399718, 9.764714959973515039167808741556, 10.35312097557229043718223000599

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