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Properties

Label 2-1078-7.2-c1-0-10
Degree $2$
Conductor $1078$
Sign $0.386 + 0.922i$
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 + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (−2 + 3.46i)5-s − 3·6-s − 0.999·8-s + (−3 + 5.19i)9-s + (1.99 + 3.46i)10-s + (0.5 + 0.866i)11-s + (−1.50 + 2.59i)12-s + 13-s + 12·15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (3 + 5.19i)18-s + (3 − 5.19i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (−0.894 + 1.54i)5-s − 1.22·6-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.632 + 1.09i)10-s + (0.150 + 0.261i)11-s + (−0.433 + 0.749i)12-s + 0.277·13-s + 3.09·15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (0.707 + 1.22i)18-s + (0.688 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056767236\)
\(L(\frac12)\) \(\approx\) \(1.056767236\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.5 + 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−10.23393613364475197564201969581, −8.781149372714475339075996803722, −7.63795437720634729569607345648, −7.14232591671462509868472146945, −6.47028032268920081019186674186, −5.67241852666744941363850489217, −4.39191030262573482839161371755, −3.14355545128427235099760877684, −2.31517844807372308535306756872, −0.837017072779915031930261662645, 0.73051834348542515273603623169, 3.58981301453723940145879756556, 4.02229385145283526546919877698, 5.02861407891087792609306280070, 5.34619257964806564440178745692, 6.31313676904364596028071653973, 7.73941977739303406073967744894, 8.392646879779355851307615976261, 9.312806924133472672576289015786, 9.769828423559958534096844007299

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