Properties

Label 2-1078-77.76-c1-0-31
Degree $2$
Conductor $1078$
Sign $0.276 + 0.961i$
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 − 3.23i·5-s + 1.84·6-s i·8-s − 0.414·9-s + 3.23·10-s + (3.28 + 0.468i)11-s + 1.84i·12-s + 6.10·13-s − 5.98·15-s + 16-s + 8.13·17-s − 0.414i·18-s − 6.61·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.06i·3-s − 0.5·4-s − 1.44i·5-s + 0.754·6-s − 0.353i·8-s − 0.138·9-s + 1.02·10-s + (0.989 + 0.141i)11-s + 0.533i·12-s + 1.69·13-s − 1.54·15-s + 0.250·16-s + 1.97·17-s − 0.0976i·18-s − 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.276 + 0.961i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.755978494\)
\(L(\frac12)\) \(\approx\) \(1.755978494\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 - 0.468i)T \)
good3 \( 1 + 1.84iT - 3T^{2} \)
5 \( 1 + 3.23iT - 5T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
17 \( 1 - 8.13T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 - 1.32iT - 29T^{2} \)
31 \( 1 + 2.35iT - 31T^{2} \)
37 \( 1 - 5.28T + 37T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 + 3.73iT - 43T^{2} \)
47 \( 1 + 1.70iT - 47T^{2} \)
53 \( 1 + 3.98T + 53T^{2} \)
59 \( 1 - 9.03iT - 59T^{2} \)
61 \( 1 + 8.55T + 61T^{2} \)
67 \( 1 - 2.35T + 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 - 4.66iT - 89T^{2} \)
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

−9.393612311521957554937930726190, −8.532864595040061057961713322103, −8.166487635383660874327760399033, −7.27612658822869057294412953465, −6.11733553488161775906305733577, −5.89481612169885596338510315533, −4.49167960486012266684047107929, −3.76336234540410931114814947964, −1.63976039107274546843277801049, −0.937319888817910696572595836168, 1.57225787955876252848544432654, 3.14896638660742600391948323132, 3.66507906881882280017693802910, 4.39440218435856123365724970480, 5.91086463455622604599460220458, 6.41072997463287943547517405856, 7.71573100208967716626415362144, 8.633461055224495656199757513237, 9.568854387973997240346928322287, 10.22015065179327982991308060662

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