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Properties

Label 2-1064-7.2-c1-0-12
Degree $2$
Conductor $1064$
Sign $0.968 - 0.250i$
Analytic cond. $8.49608$
Root an. cond. $2.91480$
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  + (−1.5 + 2.59i)5-s + (−0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)11-s + (3 + 5.19i)17-s + (0.5 − 0.866i)19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + 6·29-s + (−1 − 1.73i)31-s + (7.5 + 2.59i)35-s + 6·41-s + 9·43-s + (4.5 + 7.79i)45-s + (1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (−0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s + (0.150 + 0.261i)11-s + (0.727 + 1.26i)17-s + (0.114 − 0.198i)19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + 1.11·29-s + (−0.179 − 0.311i)31-s + (1.26 + 0.439i)35-s + 0.937·41-s + 1.37·43-s + (0.670 + 1.16i)45-s + (0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1064\)    =    \(2^{3} \cdot 7 \cdot 19\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(8.49608\)
Root analytic conductor: \(2.91480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1064} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1064,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.475846951\)
\(L(\frac12)\) \(\approx\) \(1.475846951\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.22769511629674240766279792076, −9.205301490955663687708904899765, −8.079547037625268987826141655934, −7.26290838794624075797398516820, −6.77948139946002996424791044319, −5.92142470090431956204746010342, −4.27130214828303630181140653988, −3.80196400597995206373404637535, −2.79342580269841575725242900349, −0.998838402120492877450497847243, 0.945637548330351867245348268594, 2.42428177243399102631439844589, 3.66477463995464936433010421017, 4.92954314549396733363910974375, 5.20592449328388013626858252258, 6.45706981920482044028320982333, 7.69120072171708816721776709600, 8.103981915077983012699863857576, 9.143902950389557475859413082157, 9.546790746165948065312112632774

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