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Properties

Label 2-930-93.68-c1-0-18
Degree $2$
Conductor $930$
Sign $0.999 + 0.0130i$
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 + (−1.14 + 1.29i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.29 + 1.14i)6-s + (0.742 + 1.28i)7-s + i·8-s + (−0.363 − 2.97i)9-s + (0.5 − 0.866i)10-s + (2.14 − 3.71i)11-s + (1.14 − 1.29i)12-s + (−1.46 − 0.848i)13-s + (1.28 − 0.742i)14-s + (−1.64 + 0.549i)15-s + 16-s + (2.87 + 4.97i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.662 + 0.748i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.529 + 0.468i)6-s + (0.280 + 0.486i)7-s + 0.353i·8-s + (−0.121 − 0.992i)9-s + (0.158 − 0.273i)10-s + (0.645 − 1.11i)11-s + (0.331 − 0.374i)12-s + (−0.407 − 0.235i)13-s + (0.343 − 0.198i)14-s + (−0.424 + 0.141i)15-s + 0.250·16-s + (0.697 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0130i$
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: \((2,\ 930,\ (\ :1/2),\ 0.999 + 0.0130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33549 - 0.00873854i\)
\(L(\frac12)\) \(\approx\) \(1.33549 - 0.00873854i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.29i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-2.50 - 4.97i)T \)
good7 \( 1 + (-0.742 - 1.28i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.14 + 3.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.46 + 0.848i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.87 - 4.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.59 + 2.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.16T + 23T^{2} \)
29 \( 1 + 7.18T + 29T^{2} \)
37 \( 1 + (-8.23 + 4.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.76 - 5.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.44 + 3.72i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.9iT - 47T^{2} \)
53 \( 1 + (3.03 - 5.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.18 - 0.682i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 + (-5.98 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.650 - 0.375i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.41 + 1.96i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.64 + 5.56i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.88 + 8.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.38T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−10.24576440634554357503977571951, −9.234729434767464682295534547458, −8.905527651045379293532686276465, −7.64692624210448243292998846156, −6.16428458733724038898358475923, −5.76119675907133717975810700012, −4.67876088457136837279637014452, −3.70007235625888136690475931068, −2.70724092078102977106736537163, −1.06065013968347137147409880636, 0.934967903165525612497227905449, 2.25625083815689790939573482008, 4.15905759164485198834094658017, 4.99697534687063601700163197327, 5.81109584962372114447735876419, 6.78946584648774418490876789503, 7.37863090858430527691958013812, 8.020850053629625453221549494005, 9.369053845750681593055830728457, 9.805981406514418177657086171124

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