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Properties

Label 2-1078-7.4-c1-0-5
Degree $2$
Conductor $1078$
Sign $-0.947 + 0.318i$
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  + (0.5 + 0.866i)2-s + (−0.707 + 1.22i)3-s + (−0.499 + 0.866i)4-s + (0.707 + 1.22i)5-s − 1.41·6-s − 0.999·8-s + (0.500 + 0.866i)9-s + (−0.707 + 1.22i)10-s + (0.5 − 0.866i)11-s + (−0.707 − 1.22i)12-s − 2.82·13-s − 2·15-s + (−0.5 − 0.866i)16-s + (−2.82 + 4.89i)17-s + (−0.499 + 0.866i)18-s + (1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.408 + 0.707i)3-s + (−0.249 + 0.433i)4-s + (0.316 + 0.547i)5-s − 0.577·6-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.223 + 0.387i)10-s + (0.150 − 0.261i)11-s + (−0.204 − 0.353i)12-s − 0.784·13-s − 0.516·15-s + (−0.125 − 0.216i)16-s + (−0.685 + 1.18i)17-s + (−0.117 + 0.204i)18-s + (0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.242816116\)
\(L(\frac12)\) \(\approx\) \(1.242816116\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.707 - 1.22i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (2.82 - 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.12 - 3.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.12 + 3.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (7.07 - 12.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−10.29605990375114096964612624320, −9.646495696127967982052008463983, −8.643543572871857644141732871365, −7.71481036913765700133474152413, −6.87814900858078444822581643881, −5.99810773635683109634343046609, −5.26135419763845999217246132757, −4.35159464175674344870093384405, −3.47508202460065744691953011093, −2.08252570246507862304766255653, 0.50556243447685142525612914078, 1.73414405094382841272524904422, 2.82963158174833831468337548854, 4.21907368570579354291815534310, 5.06790717818188946222630863935, 5.86302115455856576529990933302, 6.98599038472824670738269954011, 7.44423854655493122260310435010, 9.073776633135855967380822835829, 9.261872825460911455779991529314

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