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Properties

Label 2-1078-77.54-c1-0-16
Degree $2$
Conductor $1078$
Sign $0.506 - 0.862i$
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 + (0.889 − 0.513i)3-s + (0.499 + 0.866i)4-s + (−1.09 − 0.629i)5-s + 1.02·6-s + 0.999i·8-s + (−0.972 + 1.68i)9-s + (−0.629 − 1.09i)10-s + (1.45 + 2.98i)11-s + (0.889 + 0.513i)12-s + 4.08·13-s − 1.29·15-s + (−0.5 + 0.866i)16-s + (1.60 + 2.77i)17-s + (−1.68 + 0.972i)18-s + (−3.81 + 6.60i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.513 − 0.296i)3-s + (0.249 + 0.433i)4-s + (−0.487 − 0.281i)5-s + 0.419·6-s + 0.353i·8-s + (−0.324 + 0.561i)9-s + (−0.199 − 0.344i)10-s + (0.437 + 0.899i)11-s + (0.256 + 0.148i)12-s + 1.13·13-s − 0.333·15-s + (−0.125 + 0.216i)16-s + (0.388 + 0.672i)17-s + (−0.396 + 0.229i)18-s + (−0.875 + 1.51i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.506 - 0.862i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.506 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.533841550\)
\(L(\frac12)\) \(\approx\) \(2.533841550\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 2.98i)T \)
good3 \( 1 + (-0.889 + 0.513i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.09 + 0.629i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 + (-1.60 - 2.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.81 - 6.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.12 + 7.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 + (-7.95 + 4.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.154 + 0.266i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 7.57iT - 43T^{2} \)
47 \( 1 + (4.07 + 2.35i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.39 - 4.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.36 - 1.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.755 + 1.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.69 + 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.50T + 71T^{2} \)
73 \( 1 + (0.483 + 0.837i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.5 + 7.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.32T + 83T^{2} \)
89 \( 1 + (9.22 + 5.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.10526168303737011325913820616, −8.773258715221485899706545410931, −8.258071163697130179408698660920, −7.68044261722215525580252123087, −6.53549019029693169394075196627, −5.91184665434641529069162426719, −4.58485456766302936644230043897, −4.01635324278002308969859638197, −2.82382211806709539209312165750, −1.60722021518839903881329624619, 0.958291665713892427355227610811, 2.77789432174605016666116440974, 3.43027654482774087008786636629, 4.17139496272291427720501192115, 5.38285409936666967001226510598, 6.30624106462458983008553085092, 7.08634607011688850622708544704, 8.287195442842266039359883812363, 8.974404566158450001644352157919, 9.655254183733655462265906997999

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