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Properties

Label 2-930-93.26-c1-0-40
Degree $2$
Conductor $930$
Sign $-0.954 + 0.299i$
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 + (1.20 − 1.24i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−1.24 − 1.20i)6-s + (1.75 − 3.03i)7-s + i·8-s + (−0.0936 − 2.99i)9-s + (0.5 + 0.866i)10-s + (−0.508 − 0.881i)11-s + (−1.20 + 1.24i)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 + (0.695 − 0.718i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.507 − 0.492i)6-s + (0.662 − 1.14i)7-s + 0.353i·8-s + (−0.0312 − 0.999i)9-s + (0.158 + 0.273i)10-s + (−0.153 − 0.265i)11-s + (−0.347 + 0.359i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.954 + 0.299i$
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: \((2,\ 930,\ (\ :1/2),\ -0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248138 - 1.62162i\)
\(L(\frac12)\) \(\approx\) \(0.248138 - 1.62162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.24i)T \)
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

−9.832082778563116676872942504921, −8.710776111835035296226035315812, −8.079926377022116597005354157049, −7.31494170559960010171824964252, −6.53417979946029467376401706194, −5.01607610504633936656617279648, −3.99807926583649380587714149633, −3.17510746452280640986531825578, −1.96279597447566812557843263104, −0.72384967976751926294725066906, 2.00606715042190130938674632083, 3.28345596025558367229484779095, 4.41152860877919939232902038969, 5.15121625483130793360190982732, 5.94146494367273864775520705715, 7.34562089155539717491690567203, 8.149131268529000871323041190419, 8.537672362524526357520859165508, 9.504327331990074938537161078734, 10.07305529122827953514224267468

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