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Properties

Label 2-1170-13.4-c1-0-6
Degree $2$
Conductor $1170$
Sign $-0.702 - 0.711i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s + (−2.59 + 1.5i)7-s + 0.999i·8-s + (−0.5 + 0.866i)10-s + (3.23 + 1.86i)11-s + (−0.866 − 3.5i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (−1.96 + 1.13i)19-s + (−0.866 + 0.499i)20-s + (1.86 + 3.23i)22-s + (−1.73 + 3i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.981 + 0.566i)7-s + 0.353i·8-s + (−0.158 + 0.273i)10-s + (0.974 + 0.562i)11-s + (−0.240 − 0.970i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.450 + 0.260i)19-s + (−0.193 + 0.111i)20-s + (0.397 + 0.689i)22-s + (−0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.702 - 0.711i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722703657\)
\(L(\frac12)\) \(\approx\) \(1.722703657\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (0.866 + 3.5i)T \)
good7 \( 1 + (2.59 - 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.23 - 1.86i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.73 - 4.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.92iT - 31T^{2} \)
37 \( 1 + (6.86 + 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.46 + 2i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.464iT - 47T^{2} \)
53 \( 1 - 3.73T + 53T^{2} \)
59 \( 1 + (3.92 - 2.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.73 - 2.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.803 + 0.464i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 2.53iT - 83T^{2} \)
89 \( 1 + (-8.76 - 5.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 6.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.17729265771906471039062601278, −9.210392704036445613189277146992, −8.432854441467678950144891460946, −7.33532557787938922505109875687, −6.69740109799727226498089585067, −5.89985703847032863474152365346, −5.15261379048501728884108604007, −3.73511570805105454268350055854, −3.26499436897199950569818017737, −1.89377481129216352097404021403, 0.58821105687665937743756734373, 2.09264818778133787843001754403, 3.41238740382961156422863525069, 4.10509633565793349291483727393, 5.01174873928790101506290426554, 6.27131657792933937087280123357, 6.62447042026697429664124930069, 7.73507514608961743652528443472, 8.902023719453928121133528415050, 9.599239424766229911657026540358

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