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Properties

Label 2-930-155.129-c1-0-6
Degree $2$
Conductor $930$
Sign $0.909 - 0.416i$
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 + (−0.866 − 0.5i)3-s − 4-s + (−0.291 − 2.21i)5-s + (−0.5 + 0.866i)6-s + (3.11 + 1.79i)7-s + i·8-s + (0.499 + 0.866i)9-s + (−2.21 + 0.291i)10-s + (2.97 + 5.15i)11-s + (0.866 + 0.5i)12-s + (−4.69 + 2.70i)13-s + (1.79 − 3.11i)14-s + (−0.856 + 2.06i)15-s + 16-s + (−4.44 − 2.56i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 − 0.288i)3-s − 0.5·4-s + (−0.130 − 0.991i)5-s + (−0.204 + 0.353i)6-s + (1.17 + 0.679i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.701 + 0.0921i)10-s + (0.897 + 1.55i)11-s + (0.249 + 0.144i)12-s + (−1.30 + 0.751i)13-s + (0.480 − 0.831i)14-s + (−0.221 + 0.533i)15-s + 0.250·16-s + (−1.07 − 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985986 + 0.215046i\)
\(L(\frac12)\) \(\approx\) \(0.985986 + 0.215046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.291 + 2.21i)T \)
31 \( 1 + (1.89 - 5.23i)T \)
good7 \( 1 + (-3.11 - 1.79i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.97 - 5.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.69 - 2.70i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.44 + 2.56i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.31 - 4.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.06iT - 23T^{2} \)
29 \( 1 - 5.52T + 29T^{2} \)
37 \( 1 + (7.15 + 4.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.35 - 7.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.99 - 1.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.05iT - 47T^{2} \)
53 \( 1 + (-4.26 + 2.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.67 + 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 + (4.83 - 2.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.38 + 4.13i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.61 + 0.934i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.00 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.58 - 4.95i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 6.45iT - 97T^{2} \)
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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

−10.00828947434603459556150496585, −9.362724065581623696038496681285, −8.662970109342350971193805711750, −7.64672057785627115937259692690, −6.82979547026215937778693405360, −5.33820038862922887053369884785, −4.81168698675643725168231055365, −4.13352845015454578715506828510, −2.05387588867724155389915551756, −1.61005335420804980604425053908, 0.51569223976608971323686789336, 2.59974717529923019295815160643, 4.04111833135404627239190339285, 4.66108574606459015442126727679, 5.84277696439474785680482840790, 6.61747888041060705020809238573, 7.29345858428227528412822532488, 8.313000137757126696123119419669, 8.914022184437533107464516027916, 10.36507840857103263286750374258

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