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Properties

Label 2-1170-13.9-c1-0-4
Degree $2$
Conductor $1170$
Sign $-0.872 - 0.488i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s + (−1.5 − 2.59i)7-s + 0.999·8-s + (−0.5 + 0.866i)10-s + (−1.5 + 2.59i)11-s + (−2.5 + 2.59i)13-s + 3·14-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)19-s + (−0.499 − 0.866i)20-s + (−1.5 − 2.59i)22-s + (−2 + 3.46i)23-s + 25-s + (−1 − 3.46i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (−0.566 − 0.981i)7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + (−0.452 + 0.783i)11-s + (−0.693 + 0.720i)13-s + 0.801·14-s + (−0.125 + 0.216i)16-s + (−0.344 − 0.596i)19-s + (−0.111 − 0.193i)20-s + (−0.319 − 0.553i)22-s + (−0.417 + 0.722i)23-s + 0.200·25-s + (−0.196 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6098087025\)
\(L(\frac12)\) \(\approx\) \(0.6098087025\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + (2.5 - 2.59i)T \)
good7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.882328665036784916543136872324, −9.515458398083178203589082378478, −8.434411640017127906756695073347, −7.46057821440635373085044932576, −6.91243981062746642572915310007, −6.18525498011350427727088657451, −4.96714259056231123861124013442, −4.32046699485524622517912985077, −2.89024941532968279876464617694, −1.49094887698680937666205340008, 0.28899602949697007946519973347, 2.15208259862929099996698897679, 2.82868783404596387419497445391, 3.93938008579319869285337896107, 5.35632327442281067465101917507, 5.86190085978200775843700502486, 6.98389516821694913092650704392, 8.134287129670343293605283434723, 8.671764816054981294143163717384, 9.543112699505085063241803177503

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