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Properties

Label 2-930-155.109-c1-0-22
Degree $2$
Conductor $930$
Sign $0.577 + 0.816i$
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  + (0.587 + 0.809i)2-s + (−0.587 + 0.809i)3-s + (−0.309 + 0.951i)4-s + (−1.61 + 1.54i)5-s − 6-s + (−2.54 − 0.825i)7-s + (−0.951 + 0.309i)8-s + (−0.309 − 0.951i)9-s + (−2.19 − 0.401i)10-s + (1.61 − 4.95i)11-s + (−0.587 − 0.809i)12-s + (0.662 − 0.912i)13-s + (−0.825 − 2.54i)14-s + (−0.297 − 2.21i)15-s + (−0.809 − 0.587i)16-s + (−2.23 + 0.725i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.339 + 0.467i)3-s + (−0.154 + 0.475i)4-s + (−0.723 + 0.690i)5-s − 0.408·6-s + (−0.960 − 0.311i)7-s + (−0.336 + 0.109i)8-s + (−0.103 − 0.317i)9-s + (−0.695 − 0.127i)10-s + (0.485 − 1.49i)11-s + (−0.169 − 0.233i)12-s + (0.183 − 0.252i)13-s + (−0.220 − 0.678i)14-s + (−0.0768 − 0.572i)15-s + (−0.202 − 0.146i)16-s + (−0.541 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487351 - 0.252323i\)
\(L(\frac12)\) \(\approx\) \(0.487351 - 0.252323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (1.61 - 1.54i)T \)
31 \( 1 + (2.50 + 4.97i)T \)
good7 \( 1 + (2.54 + 0.825i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.61 + 4.95i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.662 + 0.912i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.23 - 0.725i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.27 - 0.923i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.529 - 0.171i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.74 + 1.99i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 6.01iT - 37T^{2} \)
41 \( 1 + (-3.89 + 2.83i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-1.15 - 1.59i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.55 + 2.14i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.80 + 2.85i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (11.3 + 8.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 8.61T + 61T^{2} \)
67 \( 1 - 4.22iT - 67T^{2} \)
71 \( 1 + (3.04 + 9.36i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (13.1 + 4.27i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.04 - 3.21i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.3 + 14.2i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.10 - 6.48i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.11 + 1.66i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.04009377811655390078034532942, −8.977190208869956407652006376197, −8.211488017747753100525824042633, −7.21854940067398188232934992175, −6.26918744123438855492105448380, −5.96019181160841472507968578673, −4.45369683171887032187391977244, −3.66078873142920003804570285644, −2.99474227412583125239031800414, −0.24227375263007237185010355504, 1.42971632823697404457342304533, 2.71869424200521025823023054187, 4.04812126411007335746012681479, 4.69105237994706952936259752675, 5.80473802634622303528863451304, 6.79746655076121244592575149134, 7.44564317833421635873233196941, 8.820240553018114308490669184692, 9.291033136636873776973018848989, 10.29729311365359864461526672498

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