Properties

Label 2-1078-77.10-c1-0-2
Degree $2$
Conductor $1078$
Sign $-0.879 - 0.475i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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  + (−0.866 + 0.5i)2-s + (−0.662 − 0.382i)3-s + (0.499 − 0.866i)4-s + (1.77 − 1.02i)5-s + 0.765·6-s + 0.999i·8-s + (−1.20 − 2.09i)9-s + (−1.02 + 1.77i)10-s + (0.649 + 3.25i)11-s + (−0.662 + 0.382i)12-s − 6.59·13-s − 1.57·15-s + (−0.5 − 0.866i)16-s + (−1.80 + 3.12i)17-s + (2.09 + 1.20i)18-s + (−0.439 − 0.760i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.382 − 0.220i)3-s + (0.249 − 0.433i)4-s + (0.794 − 0.458i)5-s + 0.312·6-s + 0.353i·8-s + (−0.402 − 0.696i)9-s + (−0.324 + 0.561i)10-s + (0.195 + 0.980i)11-s + (−0.191 + 0.110i)12-s − 1.82·13-s − 0.405·15-s + (−0.125 − 0.216i)16-s + (−0.437 + 0.758i)17-s + (0.492 + 0.284i)18-s + (−0.100 − 0.174i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.879 - 0.475i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2107350477\)
\(L(\frac12)\) \(\approx\) \(0.2107350477\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
11 \( 1 + (-0.649 - 3.25i)T \)
good3 \( 1 + (0.662 + 0.382i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.77 + 1.02i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.59T + 13T^{2} \)
17 \( 1 + (1.80 - 3.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.439 + 0.760i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.30 + 5.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.18iT - 29T^{2} \)
31 \( 1 + (5.61 + 3.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.52 - 9.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 - 7.81iT - 43T^{2} \)
47 \( 1 + (-4.97 + 2.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.214 - 0.372i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.78 + 1.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.66 + 9.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.48 - 2.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.13T + 71T^{2} \)
73 \( 1 + (5.41 - 9.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.34 + 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + (5.82 - 3.36i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.23iT - 97T^{2} \)
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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

−9.886864072584029895311543138211, −9.512202781703047959549355912305, −8.717640700403015362656416342204, −7.67895446795075772444785757059, −6.83472779179737905651854340461, −6.16902428043071384827576949181, −5.23834575846003388627022252605, −4.40094223838665528236022682339, −2.59767376682175970932933774079, −1.55823040235806519100957783765, 0.11192322501698447378999210845, 2.10504808819263050752206046603, 2.76341652958315465884051294435, 4.20205078196809299448537398504, 5.43742590161473305799648506319, 5.99238880647068750397634056734, 7.24463164734500193713100873730, 7.79745261414506203555038759211, 9.002405763295565897827128302825, 9.615827896417946094186042472034

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