Properties

Label 2-930-31.14-c1-0-14
Degree $2$
Conductor $930$
Sign $-0.883 + 0.468i$
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  + (0.309 − 0.951i)2-s + (−0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (0.499 + 0.866i)6-s + (0.422 − 4.02i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.669 + 0.743i)10-s + (−1.16 + 0.519i)11-s + (0.978 − 0.207i)12-s + (5.28 + 1.12i)13-s + (−3.69 − 1.64i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−6.00 − 2.67i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.386 + 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (0.159 − 1.52i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (0.211 + 0.235i)10-s + (−0.351 + 0.156i)11-s + (0.282 − 0.0600i)12-s + (1.46 + 0.311i)13-s + (−0.987 − 0.439i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.45 − 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209118 - 0.839977i\)
\(L(\frac12)\) \(\approx\) \(0.209118 - 0.839977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (5.05 + 2.34i)T \)
good7 \( 1 + (-0.422 + 4.02i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (1.16 - 0.519i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-5.28 - 1.12i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (6.00 + 2.67i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (-0.874 + 0.185i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (4.90 - 3.56i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.19 + 6.75i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (5.57 + 9.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.459 - 0.509i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (3.38 - 0.720i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (2.77 + 8.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.23 + 11.7i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (5.64 - 6.27i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 - 5.01T + 61T^{2} \)
67 \( 1 + (-1.65 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.153 - 1.45i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (-9.14 + 4.07i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (3.66 + 1.63i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-5.39 - 5.99i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (2.91 + 2.11i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-11.4 - 8.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.01750144432190008588552019175, −9.126869700722660976483339455114, −8.059402482683323562498445843526, −7.07951554321978758849155759496, −6.24935028380248109635565182849, −5.07158853820647651833501423307, −3.97921891219793080102086242067, −3.70926070332359338819689149152, −2.00167626826070571331747917714, −0.39469096852017620986750089954, 1.71387455242988133770816525068, 3.14631225530695340645179086830, 4.49050314321972904369797750404, 5.41114559658569401271027101797, 6.10206627061719712673044478341, 6.75982577799391725498771859653, 8.194812278805817493884677361589, 8.460282569076902055169301536290, 9.173451863275918908293078438680, 10.59694362596509967834508295379

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