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Properties

Label 2-1078-77.54-c1-0-4
Degree $2$
Conductor $1078$
Sign $-0.968 - 0.248i$
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 + (−0.937 + 0.541i)3-s + (0.499 + 0.866i)4-s + (−0.937 − 0.541i)5-s − 1.08·6-s + 0.999i·8-s + (−0.914 + 1.58i)9-s + (−0.541 − 0.937i)10-s + (3.30 − 0.275i)11-s + (−0.937 − 0.541i)12-s − 2.29·13-s + 1.17·15-s + (−0.5 + 0.866i)16-s + (−1.58 + 0.914i)18-s + (−2.77 + 4.80i)19-s − 1.08i·20-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.541 + 0.312i)3-s + (0.249 + 0.433i)4-s + (−0.419 − 0.242i)5-s − 0.441·6-s + 0.353i·8-s + (−0.304 + 0.527i)9-s + (−0.171 − 0.296i)10-s + (0.996 − 0.0829i)11-s + (−0.270 − 0.156i)12-s − 0.636·13-s + 0.302·15-s + (−0.125 + 0.216i)16-s + (−0.373 + 0.215i)18-s + (−0.635 + 1.10i)19-s − 0.242i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.968 - 0.248i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9674365843\)
\(L(\frac12)\) \(\approx\) \(0.9674365843\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 + 0.275i)T \)
good3 \( 1 + (0.937 - 0.541i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.937 + 0.541i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.29T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.77 - 4.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + (1.21 - 0.699i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (4.25 + 2.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.24 + 3.88i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.23 - 3.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.77 - 4.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.82 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-7.83 - 13.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (-3.08 - 1.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.60iT - 97T^{2} \)
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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

−10.33417100877002507072995448876, −9.507719924209536790879555760768, −8.343751337803249722659448184272, −7.82382390534458707542381105693, −6.69617650818489457955560461569, −5.94104367816233292292365204003, −5.07572906507861529306108301517, −4.27695315049996654838191850864, −3.41166279328520347672699885071, −1.86892904280189085946006292145, 0.35859195554912498422489867606, 1.98056213336998801781794960353, 3.25720233942363294253648157335, 4.17601852116866628880960367152, 5.09987604338212833742553369143, 6.22622561407853717795817876518, 6.69578647183428836743542935078, 7.59749929419030146533086384151, 8.854517652897713222912416226294, 9.515690022321698925200678976376

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