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Properties

Label 2-930-93.26-c1-0-27
Degree $2$
Conductor $930$
Sign $0.699 - 0.714i$
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 + (−0.474 + 1.66i)3-s − 4-s + (0.866 − 0.5i)5-s + (−1.66 − 0.474i)6-s + (1.75 − 3.03i)7-s − i·8-s + (−2.54 − 1.58i)9-s + (0.5 + 0.866i)10-s + (0.508 + 0.881i)11-s + (0.474 − 1.66i)12-s + (−0.482 + 0.278i)13-s + (3.03 + 1.75i)14-s + (0.422 + 1.67i)15-s + 16-s + (0.748 − 1.29i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.273 + 0.961i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.680 − 0.193i)6-s + (0.662 − 1.14i)7-s − 0.353i·8-s + (−0.849 − 0.526i)9-s + (0.158 + 0.273i)10-s + (0.153 + 0.265i)11-s + (0.136 − 0.480i)12-s + (−0.133 + 0.0772i)13-s + (0.811 + 0.468i)14-s + (0.108 + 0.433i)15-s + 0.250·16-s + (0.181 − 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.699 - 0.714i$
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: \((2,\ 930,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41817 + 0.596300i\)
\(L(\frac12)\) \(\approx\) \(1.41817 + 0.596300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.474 - 1.66i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-5.39 - 1.39i)T \)
good7 \( 1 + (-1.75 + 3.03i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.508 - 0.881i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.482 - 0.278i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.748 + 1.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.58 + 4.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.21T + 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
37 \( 1 + (-2.08 - 1.20i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.86 + 4.54i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.69 + 3.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.95iT - 47T^{2} \)
53 \( 1 + (-1.80 - 3.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.22 + 4.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 0.662iT - 61T^{2} \)
67 \( 1 + (-0.768 - 1.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.71 - 5.61i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.23 - 3.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.184 - 0.106i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.71 - 2.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.779T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−10.05252447120408037668141692059, −9.371351236032851074686265534204, −8.588849891949963182436910670281, −7.55548298416758392538692956923, −6.78900585014045736715121948016, −5.72613975998154321707706014172, −4.73329376660031400465182164398, −4.40100070589620059820519761722, −3.02262774256732607452194602851, −0.921361444941150109620751038217, 1.25389626053282937927238934447, 2.25289643681787595143141287557, 3.16269747760785092245864349228, 4.80520742349628765310236351645, 5.68040231810177954566842080919, 6.31476175845522311805214272666, 7.61146690205508311211800842788, 8.328383351864991406688819452255, 9.070953524829885191488624427461, 10.08968813082460149206039668180

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