Properties

Label 2-930-31.25-c1-0-18
Degree $2$
Conductor $930$
Sign $0.695 + 0.718i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−2 − 3.46i)7-s + 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (1.5 − 2.59i)11-s + (0.5 + 0.866i)12-s + (1 − 1.73i)13-s + (−2 − 3.46i)14-s + 0.999·15-s + 16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.755 − 1.30i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (0.452 − 0.783i)11-s + (0.144 + 0.249i)12-s + (0.277 − 0.480i)13-s + (−0.534 − 0.925i)14-s + 0.258·15-s + 0.250·16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35713 - 0.999707i\)
\(L(\frac12)\) \(\approx\) \(2.35713 - 0.999707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-2 - 5.19i)T \)
good7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + T + 47T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.09525906558232194306946797242, −9.199584065793971541431103152329, −8.341615241896789218493879833880, −7.22425382334881124565689590245, −6.49775466888027641132114124234, −5.47668710373780380993700138658, −4.48950715216738713579792303945, −3.66446007861820986766826500574, −2.86405044815947304372246796157, −0.968926216923291339214874450952, 1.86724401278316700695794125397, 2.69008970981109171078380846317, 3.72833159292597274449965840144, 4.93861677558081285453540997981, 6.15496752736418220895976880189, 6.43982957829245950124572615205, 7.42279795918684088903537232660, 8.569516861440567197812080845400, 9.303161367808781556979367266984, 10.09614873018416241204703557101

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