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Properties

Label 2-1530-85.4-c1-0-1
Degree $2$
Conductor $1530$
Sign $-0.429 - 0.902i$
Analytic cond. $12.2171$
Root an. cond. $3.49529$
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  − 2-s + 4-s + (−0.707 − 2.12i)5-s + (−1 + i)7-s − 8-s + (0.707 + 2.12i)10-s + (1.58 + 1.58i)11-s + 3i·13-s + (1 − i)14-s + 16-s + (−2.12 + 3.53i)17-s − 7.24i·19-s + (−0.707 − 2.12i)20-s + (−1.58 − 1.58i)22-s + (2.82 − 2.82i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.316 − 0.948i)5-s + (−0.377 + 0.377i)7-s − 0.353·8-s + (0.223 + 0.670i)10-s + (0.478 + 0.478i)11-s + 0.832i·13-s + (0.267 − 0.267i)14-s + 0.250·16-s + (−0.514 + 0.857i)17-s − 1.66i·19-s + (−0.158 − 0.474i)20-s + (−0.338 − 0.338i)22-s + (0.589 − 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1530\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.429 - 0.902i$
Analytic conductor: \(12.2171\)
Root analytic conductor: \(3.49529\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1530} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1530,\ (\ :1/2),\ -0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4350029149\)
\(L(\frac12)\) \(\approx\) \(0.4350029149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (0.707 + 2.12i)T \)
17 \( 1 + (2.12 - 3.53i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-1.58 - 1.58i)T + 11iT^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
19 \( 1 + 7.24iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T - 29iT^{2} \)
31 \( 1 + (5.36 - 5.36i)T - 31iT^{2} \)
37 \( 1 + (5.24 + 5.24i)T + 37iT^{2} \)
41 \( 1 + (4.41 + 4.41i)T + 41iT^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 - 1.58iT - 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 + (-6.12 - 6.12i)T + 61iT^{2} \)
67 \( 1 - 14.4iT - 67T^{2} \)
71 \( 1 + (3.70 - 3.70i)T - 71iT^{2} \)
73 \( 1 + (-8.36 - 8.36i)T + 73iT^{2} \)
79 \( 1 + (0.242 + 0.242i)T + 79iT^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (0.121 + 0.121i)T + 97iT^{2} \)
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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

−9.370039322224895420371125637709, −8.882950312687063933238533067684, −8.545695849730762849721450538242, −7.15893921004094896693616939309, −6.83368398391713486032756861751, −5.64499697771399175968692644996, −4.69740855858344061264219942069, −3.81436070329337352956421235736, −2.42711422191024531241739799090, −1.32767140543724009150106020858, 0.22205672631529043725315886261, 1.84003841597120701716097332832, 3.23277689372951457013509767710, 3.64005994430510314146666789273, 5.20576324354879690772173685423, 6.26353034202639752268574225099, 6.82625708084552584929854119902, 7.72656790443981188605870614757, 8.228015303525287603249783533813, 9.378275610625672124650841191791

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