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Properties

Label 2-930-93.26-c1-0-13
Degree $2$
Conductor $930$
Sign $0.798 - 0.602i$
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 + 1.5i)3-s − 4-s + (0.866 − 0.5i)5-s + (1.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + i·8-s + (−1.5 − 2.59i)9-s + (−0.5 − 0.866i)10-s + (2.59 + 4.5i)11-s + (0.866 − 1.5i)12-s + (−3 + 1.73i)13-s + (−0.866 − 0.5i)14-s + 1.73i·15-s + 16-s + (1.73 − 3i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 + 0.866i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (0.612 + 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.158 − 0.273i)10-s + (0.783 + 1.35i)11-s + (0.249 − 0.433i)12-s + (−0.832 + 0.480i)13-s + (−0.231 − 0.133i)14-s + 0.447i·15-s + 0.250·16-s + (0.420 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.798 - 0.602i$
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: \((2,\ 930,\ (\ :1/2),\ 0.798 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18536 + 0.396814i\)
\(L(\frac12)\) \(\approx\) \(1.18536 + 0.396814i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (5.5 + 0.866i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
37 \( 1 + (-6 - 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.19 - 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 - 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + (-6.06 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.9 - 7.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.3 + 6i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9 + 5.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.59 + 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.09013298850589249216522162241, −9.461630450335772226784289760020, −9.072358009457933264383642999417, −7.61043741709646336707270447917, −6.68616218034412165322903133104, −5.49180064158375595579167208877, −4.63515798158073784367472199683, −4.08815879434775405704147675419, −2.71598343930472018465862540324, −1.30504364245393766772425949845, 0.71439241652872735583149676646, 2.28910609151259463945568030526, 3.64273198799905050629209685702, 5.31518518774862399134446897733, 5.57038186957210885266663455092, 6.72153880524251968358724154211, 7.11886905820432567935058806394, 8.361909944957557368662587049580, 8.762245546092680903156865186510, 9.929131486238761481741222296455

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