Properties

Label 2-930-31.5-c1-0-13
Degree $2$
Conductor $930$
Sign $0.848 + 0.528i$
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 + (1.5 − 2.59i)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 + (1.5 − 2.59i)14-s + 0.999·15-s + 16-s + (2 − 3.46i)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 + (0.566 − 0.981i)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 + (0.400 − 0.694i)14-s + 0.258·15-s + 0.250·16-s + (0.485 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.848 + 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91204 - 0.833119i\)
\(L(\frac12)\) \(\approx\) \(2.91204 - 0.833119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.5 - 0.866i)T \)
good7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
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 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4 - 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.06313336648752984205782750850, −9.259186178159370207996579597284, −7.976943537951356950966702021411, −7.31338212734660411489206356737, −6.69896880412838919509821790879, −5.68280636162624683313065548242, −4.50015076232325173498030338145, −3.76954057489232609354877021191, −2.46838012742712737135329819441, −1.36320197114878326546102890200, 1.62380532765996854711342145857, 2.87811219297513495013808418306, 3.87030646276061031301297979428, 4.86889098220833586022263647804, 5.75902762572781259140433119921, 6.26054626222910571322493439985, 7.929063340094521089099421073307, 8.355019401360742023099072822766, 9.291772934936483037937818557628, 10.15490639733422592398993750236

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