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Properties

Label 2-1078-7.4-c1-0-11
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 + (−1.41 + 2.44i)3-s + (−0.499 + 0.866i)4-s + 2.82·6-s + 0.999·8-s + (−2.49 − 4.33i)9-s + (0.5 − 0.866i)11-s + (−1.41 − 2.44i)12-s + 4.24·13-s + (−0.5 − 0.866i)16-s + (1.41 − 2.44i)17-s + (−2.5 + 4.33i)18-s + (2.12 + 3.67i)19-s − 0.999·22-s + (−3 − 5.19i)23-s + (−1.41 + 2.44i)24-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.816 + 1.41i)3-s + (−0.249 + 0.433i)4-s + 1.15·6-s + 0.353·8-s + (−0.833 − 1.44i)9-s + (0.150 − 0.261i)11-s + (−0.408 − 0.707i)12-s + 1.17·13-s + (−0.125 − 0.216i)16-s + (0.342 − 0.594i)17-s + (−0.589 + 1.02i)18-s + (0.486 + 0.842i)19-s − 0.213·22-s + (−0.625 − 1.08i)23-s + (−0.288 + 0.499i)24-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

\(L(1)\) \(\approx\) \(0.9919968701\)
\(L(\frac12)\) \(\approx\) \(0.9919968701\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.12 - 3.67i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-3.53 + 6.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-6.36 - 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.65 - 9.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-4.24 + 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−10.04782721068340521854276309270, −9.356936636813252356059572395977, −8.625616295952235757414077206119, −7.64052900021886298656851426515, −6.16565291277193775863964397411, −5.65331695370207060271284626456, −4.36382061921744176456898733077, −3.93916196950605438771150927825, −2.74174831006288833916842200838, −0.835246760862546430766975005413, 0.905456057023445651646997909672, 1.85187184812690158034404391217, 3.60119681789291084263510779006, 5.13649040754469950502269705972, 5.81593142170049109768435747105, 6.58264945385010845091290977510, 7.23578761412582433242552160222, 7.940360557518477234317406468495, 8.784229410582820689897826088916, 9.714211581038556159289075838540

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