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Properties

Label 2-930-155.37-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.800 + 0.599i$
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  + (0.707 + 0.707i)2-s + (−0.258 + 0.965i)3-s + 1.00i·4-s + (−2.06 − 0.848i)5-s + (−0.866 + 0.500i)6-s + (0.339 − 1.26i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.862 − 2.06i)10-s + (4.94 + 2.85i)11-s + (−0.965 − 0.258i)12-s + (−6.23 + 1.67i)13-s + (1.13 − 0.655i)14-s + (1.35 − 1.77i)15-s − 1.00·16-s + (−6.57 − 1.76i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.925 − 0.379i)5-s + (−0.353 + 0.204i)6-s + (0.128 − 0.478i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.272 − 0.652i)10-s + (1.48 + 0.860i)11-s + (−0.278 − 0.0747i)12-s + (−1.72 + 0.463i)13-s + (0.303 − 0.175i)14-s + (0.349 − 0.459i)15-s − 0.250·16-s + (−1.59 − 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.800 + 0.599i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111186 - 0.333749i\)
\(L(\frac12)\) \(\approx\) \(0.111186 - 0.333749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.06 + 0.848i)T \)
31 \( 1 + (-4.04 + 3.82i)T \)
good7 \( 1 + (-0.339 + 1.26i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.94 - 2.85i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.23 - 1.67i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (6.57 + 1.76i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.15 - 2.40i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.66 + 1.66i)T + 23iT^{2} \)
29 \( 1 + 7.52T + 29T^{2} \)
37 \( 1 + (2.36 + 0.634i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.748 + 1.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.59 + 5.93i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.556 + 0.556i)T + 47iT^{2} \)
53 \( 1 + (11.3 - 3.03i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (8.97 - 5.18i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.06iT - 61T^{2} \)
67 \( 1 + (-7.40 + 1.98i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.75 - 11.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.69 + 2.59i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.40 - 7.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.1 + 2.71i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + (5.21 + 5.21i)T + 97iT^{2} \)
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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

−10.67874661781250317356922019750, −9.463629569646452180778075072631, −9.038380968304122209432106550976, −7.86637739601954184102453202431, −7.09070210477644310586079657705, −6.45724863739906099723586429905, −4.99987371427476783030793926350, −4.26590200971317492093607484726, −3.99148121620432961526302522727, −2.20886253066816381731299174358, 0.13344403278194952202083641727, 1.95542411233886546588922491186, 3.01662451548509801046991686630, 4.11340704755020801798939077120, 4.96803913456196372451253841724, 6.28383199594749247486607505118, 6.77217246241583399556241089350, 7.84657910395065009513467875240, 8.753213759332995845930299391349, 9.529698378490685298193603595672

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