Properties

Label 2-930-93.68-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.994 + 0.100i$
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 + (−1.69 − 0.338i)3-s − 4-s + (−0.866 − 0.5i)5-s + (0.338 − 1.69i)6-s + (−1.24 − 2.15i)7-s i·8-s + (2.77 + 1.14i)9-s + (0.5 − 0.866i)10-s + (0.404 − 0.700i)11-s + (1.69 + 0.338i)12-s + (3.21 + 1.85i)13-s + (2.15 − 1.24i)14-s + (1.30 + 1.14i)15-s + 16-s + (3.37 + 5.84i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.980 − 0.195i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.138 − 0.693i)6-s + (−0.471 − 0.816i)7-s − 0.353i·8-s + (0.923 + 0.383i)9-s + (0.158 − 0.273i)10-s + (0.121 − 0.211i)11-s + (0.490 + 0.0976i)12-s + (0.892 + 0.515i)13-s + (0.577 − 0.333i)14-s + (0.336 + 0.294i)15-s + 0.250·16-s + (0.818 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00603261 - 0.119914i\)
\(L(\frac12)\) \(\approx\) \(0.00603261 - 0.119914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.338i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (5.56 - 0.290i)T \)
good7 \( 1 + (1.24 + 2.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.404 + 0.700i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.21 - 1.85i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.37 - 5.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.81T + 23T^{2} \)
29 \( 1 + 9.79T + 29T^{2} \)
37 \( 1 + (3.94 - 2.27i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.77 + 1.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.26 + 3.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.3iT - 47T^{2} \)
53 \( 1 + (4.33 - 7.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.80 - 1.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.93iT - 61T^{2} \)
67 \( 1 + (-1.82 + 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.69 - 3.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.0 + 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.86 - 2.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.60 - 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.63T + 89T^{2} \)
97 \( 1 - 3.29T + 97T^{2} \)
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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

−10.76409950736483278808686109776, −9.615913892303715735244753648318, −8.734528303072569730510525565946, −7.72963223601347854285387982077, −7.01892538494433300734880445610, −6.23017844365371832918650498630, −5.54664600679030465693028671358, −4.27732124488005640915020286090, −3.75536236319458639008802803529, −1.41792210951410602799499784715, 0.06845115437951351135604066046, 1.78133393726405493430798922796, 3.32469351725496245957512305434, 4.04137049852495879670851199761, 5.42783190791324412389682853029, 5.82568537832358377220696525249, 7.01362617029277465738553154179, 8.004485598945265812312638247797, 9.100727582896705125091892712139, 9.828677124301302408311813662653

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