Properties

Label 2-1170-13.3-c1-0-15
Degree $2$
Conductor $1170$
Sign $0.128 + 0.991i$
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 + (−1.78 + 3.08i)7-s − 0.999·8-s + (−0.5 − 0.866i)10-s + (−2.06 − 3.57i)11-s + (−3.34 + 1.35i)13-s − 3.56·14-s + (−0.5 − 0.866i)16-s + (2.56 − 4.43i)17-s + (1.78 − 3.08i)19-s + (0.499 − 0.866i)20-s + (2.06 − 3.57i)22-s + (−3.84 − 6.65i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.673 + 1.16i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (−0.621 − 1.07i)11-s + (−0.926 + 0.375i)13-s − 0.951·14-s + (−0.125 − 0.216i)16-s + (0.621 − 1.07i)17-s + (0.408 − 0.707i)19-s + (0.111 − 0.193i)20-s + (0.439 − 0.761i)22-s + (−0.801 − 1.38i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.128 + 0.991i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4562775276\)
\(L(\frac12)\) \(\approx\) \(0.4562775276\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.34 - 1.35i)T \)
good7 \( 1 + (1.78 - 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.56 + 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.84 + 6.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.28 - 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + (2.06 + 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 - 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (-5.28 + 9.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.12 + 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.43 - 4.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 7.43T + 79T^{2} \)
83 \( 1 + 1.12T + 83T^{2} \)
89 \( 1 + (-0.903 - 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.561 - 0.972i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.389229176577256544261690626565, −8.665270233846152077282399999969, −7.968188571998846357872101841031, −6.98598385497178588405945249553, −6.24839984934933855310698617700, −5.31184017457850902091929999632, −4.67313871087248402472005420313, −3.17559784853414327900779061942, −2.66605665155458832888053097825, −0.17336768521322871030647396929, 1.48234768449722870925741437243, 2.90947819592377417424378958648, 3.81770032688827370200436619814, 4.55822160272687823846799248988, 5.59013702407048418358423884709, 6.66728272190647394286437219377, 7.62417105331064707958935710392, 8.074337186320830070563668062788, 9.649278582196758825981679064032, 10.16005257660228774141216839918

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