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Properties

Label 2-930-93.26-c1-0-31
Degree $2$
Conductor $930$
Sign $0.277 + 0.960i$
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 + (0.118 + 1.72i)3-s − 4-s + (0.866 − 0.5i)5-s + (1.72 − 0.118i)6-s + (0.295 − 0.511i)7-s + i·8-s + (−2.97 + 0.408i)9-s + (−0.5 − 0.866i)10-s + (−2.33 − 4.05i)11-s + (−0.118 − 1.72i)12-s + (2.48 − 1.43i)13-s + (−0.511 − 0.295i)14-s + (0.966 + 1.43i)15-s + 16-s + (1.50 − 2.60i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.0681 + 0.997i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (0.705 − 0.0482i)6-s + (0.111 − 0.193i)7-s + 0.353i·8-s + (−0.990 + 0.136i)9-s + (−0.158 − 0.273i)10-s + (−0.705 − 1.22i)11-s + (−0.0340 − 0.498i)12-s + (0.690 − 0.398i)13-s + (−0.136 − 0.0788i)14-s + (0.249 + 0.371i)15-s + 0.250·16-s + (0.364 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15767 - 0.870976i\)
\(L(\frac12)\) \(\approx\) \(1.15767 - 0.870976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.118 - 1.72i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-4.99 + 2.45i)T \)
good7 \( 1 + (-0.295 + 0.511i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.33 + 4.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.48 + 1.43i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.50 + 2.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.44 + 2.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.585T + 23T^{2} \)
29 \( 1 + 5.47T + 29T^{2} \)
37 \( 1 + (-4.88 - 2.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.52 + 2.61i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.86 - 1.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.89iT - 47T^{2} \)
53 \( 1 + (-0.900 - 1.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.70 - 1.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.09iT - 61T^{2} \)
67 \( 1 + (5.77 + 9.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.46 - 0.844i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.55 + 1.47i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.835 - 0.482i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.409T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.936838395832918664793107431284, −9.244327044870854676968521138235, −8.511237764808853543302463383202, −7.69924084915413149870561790850, −6.01381287288244363051795073138, −5.40769266629588118910118821866, −4.45810990149779788890430297951, −3.38356307420326758552562089879, −2.62676382017802175172062197759, −0.72955990538475291092794126789, 1.45957645173743789247305719255, 2.61726532636875456868998098226, 4.05869191404630962912211842636, 5.37206859624570017542515506635, 6.01702872023073073656253396622, 6.90925505836190025559674452118, 7.62167769326797131667567194467, 8.304189122270332809385042887215, 9.245249419196982818615023811183, 10.08437274988412976135442424018

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