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Properties

Label 2-1170-39.2-c1-0-10
Degree $2$
Conductor $1170$
Sign $0.976 + 0.213i$
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 + (−2.36 + 0.633i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.500i)10-s + (−0.965 − 0.258i)11-s + (1 − 3.46i)13-s − 2.44i·14-s + (0.500 + 0.866i)16-s + (2.63 − 4.57i)17-s + (−1 − 3.73i)19-s + (−0.258 − 0.965i)20-s + (0.499 − 0.866i)22-s + (0.258 + 0.448i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.316 + 0.316i)5-s + (−0.894 + 0.239i)7-s + (0.249 − 0.249i)8-s + (−0.273 + 0.158i)10-s + (−0.291 − 0.0780i)11-s + (0.277 − 0.960i)13-s − 0.654i·14-s + (0.125 + 0.216i)16-s + (0.640 − 1.10i)17-s + (−0.229 − 0.856i)19-s + (−0.0578 − 0.215i)20-s + (0.106 − 0.184i)22-s + (0.0539 + 0.0934i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.976 + 0.213i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.976 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.157619628\)
\(L(\frac12)\) \(\approx\) \(1.157619628\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good7 \( 1 + (2.36 - 0.633i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.63 + 4.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.258 - 0.448i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.03 + 4.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.09 - 7.09i)T - 31iT^{2} \)
37 \( 1 + (0.133 - 0.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.82 + 10.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.7 - 6.23i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.53 + 3.53i)T - 47iT^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 + (3.55 + 13.2i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.83 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.09 + 1.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-14.0 + 3.77i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (7.92 + 7.92i)T + 73iT^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + (-7.72 - 7.72i)T + 83iT^{2} \)
89 \( 1 + (9.84 + 2.63i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.464 - 1.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.551624500942894931915534876422, −9.049374176215976820716189158537, −8.000572397823500963847776445286, −7.22614528039831453699890969560, −6.43421485492556709088888514921, −5.65908335717142907880199453672, −4.87348646738726158439123041323, −3.43685101516824253952555604146, −2.59481783484662745805284124011, −0.60565922673695509286986086950, 1.21561074317202994219480579972, 2.41364267862854899892246212888, 3.62967193322472324491662854262, 4.32091151758447702767202932101, 5.63175300810343682752076881917, 6.36536949033736905609209828327, 7.43383600811299454133484291922, 8.403786352456028286334893426976, 9.126446633058683493271085128179, 9.921762399117906942473945138417

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