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Properties

Label 2-1170-9.4-c1-0-40
Degree $2$
Conductor $1170$
Sign $-0.766 + 0.642i$
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 + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.5 − 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s − 0.999·10-s + (−0.5 − 0.866i)11-s + 1.73i·12-s + (0.5 − 0.866i)13-s + (−0.999 + 1.73i)14-s − 1.73i·15-s + (−0.5 − 0.866i)16-s + 5·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.612 − 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s − 0.316·10-s + (−0.150 − 0.261i)11-s + 0.499i·12-s + (0.138 − 0.240i)13-s + (−0.267 + 0.462i)14-s − 0.447i·15-s + (−0.125 − 0.216i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.720981154\)
\(L(\frac12)\) \(\approx\) \(1.720981154\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.542733364060192697479540794926, −8.705762549498479129573308356160, −7.88181439842747759055738872485, −7.33639120206070033381945123611, −6.23337387612351573109090487188, −5.06591458243817935201487163011, −3.70789082175443550073615353591, −3.19682486438564950890669974157, −1.86347692023748648719539082093, −0.77997820214502987274995757428, 1.76990634322557225197324702088, 2.98278483395025105590938094060, 3.86820911471094054670068148027, 5.22731937787428209721603290473, 5.77299585749420895778709015082, 7.13771498889159458550506118884, 7.51040352647099975014178989990, 8.728944802199760953130258409130, 9.050189613719278429468195452912, 10.04813052320260542104036511816

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