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Properties

Label 2-1078-77.76-c1-0-2
Degree $2$
Conductor $1078$
Sign $-0.975 - 0.220i$
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  + i·2-s − 1.84i·3-s − 4-s + 2.47i·5-s + 1.84·6-s − i·8-s − 0.414·9-s − 2.47·10-s + (−1.99 + 2.65i)11-s + 1.84i·12-s − 1.96·13-s + 4.56·15-s + 16-s − 5.64·17-s − 0.414i·18-s − 0.906·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.06i·3-s − 0.5·4-s + 1.10i·5-s + 0.754·6-s − 0.353i·8-s − 0.138·9-s − 0.781·10-s + (−0.600 + 0.799i)11-s + 0.533i·12-s − 0.544·13-s + 1.17·15-s + 0.250·16-s − 1.37·17-s − 0.0976i·18-s − 0.208·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5529864198\)
\(L(\frac12)\) \(\approx\) \(0.5529864198\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (1.99 - 2.65i)T \)
good3 \( 1 + 1.84iT - 3T^{2} \)
5 \( 1 - 2.47iT - 5T^{2} \)
13 \( 1 + 1.96T + 13T^{2} \)
17 \( 1 + 5.64T + 17T^{2} \)
19 \( 1 + 0.906T + 19T^{2} \)
23 \( 1 + 0.936T + 23T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + 4.71iT - 31T^{2} \)
37 \( 1 + 9.62T + 37T^{2} \)
41 \( 1 + 6.93T + 41T^{2} \)
43 \( 1 - 6.80iT - 43T^{2} \)
47 \( 1 - 4.00iT - 47T^{2} \)
53 \( 1 - 6.56T + 53T^{2} \)
59 \( 1 - 0.965iT - 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 6.72T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 4.83T + 83T^{2} \)
89 \( 1 - 12.7iT - 89T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−10.28970611143607402829500817270, −9.343754591975154561212098650183, −8.291449591672524857511765717105, −7.49496770633155651228661593199, −6.84937277207358911242008125445, −6.56715051989116819448812742253, −5.29434908872864358868204690524, −4.27892470381740216414489146407, −2.84160067635662428198233525117, −1.85755210577548172187197347877, 0.23227231093499160024724149523, 1.94426881576590211043873490794, 3.28112150755361367271732000385, 4.28949438100872138069024645797, 4.87669695424763474146467610709, 5.62255641541956085264732501523, 7.02653883339521184005466552934, 8.420748513047071636662337320891, 8.752482553676883533396921028725, 9.582413983761425474481194830010

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