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Properties

Label 2-930-93.26-c1-0-34
Degree $2$
Conductor $930$
Sign $0.998 - 0.0600i$
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 + (1.68 + 0.417i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.417 + 1.68i)6-s + (2.16 − 3.75i)7-s − i·8-s + (2.65 + 1.40i)9-s + (−0.5 − 0.866i)10-s + (−2.88 − 5.00i)11-s + (−1.68 − 0.417i)12-s + (0.829 − 0.478i)13-s + (3.75 + 2.16i)14-s + (−1.66 + 0.478i)15-s + 16-s + (0.169 − 0.293i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.970 + 0.241i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.170 + 0.686i)6-s + (0.819 − 1.42i)7-s − 0.353i·8-s + (0.883 + 0.467i)9-s + (−0.158 − 0.273i)10-s + (−0.871 − 1.50i)11-s + (−0.485 − 0.120i)12-s + (0.230 − 0.132i)13-s + (1.00 + 0.579i)14-s + (−0.429 + 0.123i)15-s + 0.250·16-s + (0.0410 − 0.0710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.998 - 0.0600i$
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: \((2,\ 930,\ (\ :1/2),\ 0.998 - 0.0600i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06508 + 0.0621032i\)
\(L(\frac12)\) \(\approx\) \(2.06508 + 0.0621032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.417i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-4.75 + 2.90i)T \)
good7 \( 1 + (-2.16 + 3.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.88 + 5.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.829 + 0.478i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.169 + 0.293i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.915 + 1.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 + 7.24T + 29T^{2} \)
37 \( 1 + (5.82 + 3.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.47 - 0.849i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 2.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.77iT - 47T^{2} \)
53 \( 1 + (-4.01 - 6.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.461 + 0.266i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 + (-6.77 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.72 - 3.30i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.51 + 3.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 + 1.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.17 + 7.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.143T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.06552398093598254152793067831, −8.883174735219497571433059076452, −8.336046833457820525932093540099, −7.48367047348312517667811260747, −7.16408026337316001417580902367, −5.68056992171739164727717214104, −4.66633339596869491852459422820, −3.80985976212546986370890280612, −2.91765583248643841074145442709, −0.958794621588513904283024735233, 1.67008669095782791381653216334, 2.38590957198592543065311481347, 3.46030925939509032336032397442, 4.72754103913041924528214060748, 5.29497123299466126859427821030, 6.92311382186255517616988072074, 7.84240006682460702838372570822, 8.484748382909664456599084430993, 9.181375208336867758037015828923, 9.896311953786696309946007577747

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