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Properties

Label 2-1170-39.20-c1-0-13
Degree $2$
Conductor $1170$
Sign $0.388 + 0.921i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.866 − 0.232i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.500i)10-s + (−0.965 + 0.258i)11-s + (−2.59 − 2.5i)13-s − 0.896i·14-s + (0.500 − 0.866i)16-s + (−2.44 − 4.24i)17-s + (0.232 − 0.866i)19-s + (−0.258 + 0.965i)20-s + (−0.499 − 0.866i)22-s + (2.63 − 4.57i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.316 − 0.316i)5-s + (−0.327 − 0.0877i)7-s + (−0.249 − 0.249i)8-s + (0.273 + 0.158i)10-s + (−0.291 + 0.0780i)11-s + (−0.720 − 0.693i)13-s − 0.239i·14-s + (0.125 − 0.216i)16-s + (−0.594 − 1.02i)17-s + (0.0532 − 0.198i)19-s + (−0.0578 + 0.215i)20-s + (−0.106 − 0.184i)22-s + (0.550 − 0.953i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.027507690\)
\(L(\frac12)\) \(\approx\) \(1.027507690\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good7 \( 1 + (0.866 + 0.232i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.44 + 4.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.232 + 0.866i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.63 + 4.57i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.26 - 1.26i)T + 31iT^{2} \)
37 \( 1 + (-1.59 - 5.96i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.03 + 3.86i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.92 - 2.26i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.02 + 7.02i)T + 47iT^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + (-1.03 + 3.86i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.46 - 4.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.1 + 2.73i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.93 - 0.517i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.53 + 2.53i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (-9.52 + 9.52i)T - 83iT^{2} \)
89 \( 1 + (11.7 - 3.13i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.66 - 6.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.643804235080709499889399038309, −8.680137044015451533831955078516, −7.984589915753918239061277236250, −6.97930177193466068526821605567, −6.43797095264475979717636302359, −5.13182241991186963500595818250, −4.88530256732026198416636938758, −3.44991215091854074979777158412, −2.37060828992486364573630886950, −0.40369357287719747627117214704, 1.62400619305524187681290728573, 2.64282491429463447822323653816, 3.66389136607403287497133318259, 4.65468792339634371474854374387, 5.65325011266056096182916271893, 6.49591037824492528611099287755, 7.44003499953301435957599553081, 8.459969549965290799152512872682, 9.446661794318227365736675004057, 9.823159530392605193247965037296

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