Properties

Label 2-1078-7.2-c1-0-3
Degree $2$
Conductor $1078$
Sign $0.605 - 0.795i$
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 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s − 1.99·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.999 + 1.73i)10-s + (−0.5 − 0.866i)11-s + (−0.999 + 1.73i)12-s − 4·13-s + 3.99·15-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (−2 + 3.46i)19-s + 1.99·20-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.577 − 0.999i)3-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s − 0.816·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.288 + 0.499i)12-s − 1.10·13-s + 1.03·15-s + (−0.125 + 0.216i)16-s + (0.117 + 0.204i)18-s + (−0.458 + 0.794i)19-s + 0.447·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.605 - 0.795i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4611107178\)
\(L(\frac12)\) \(\approx\) \(0.4611107178\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7 - 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.24803667058770397795027370933, −9.393050801438464190609356132860, −8.114974726979999944136498737233, −7.37428265443749302439275153552, −6.61994063891904614593524159116, −5.84619022743674287554604338135, −4.80258119780136087905492633840, −3.61307303972398834293074101973, −2.62900969894857450786351264314, −1.40038414670975303335191805557, 0.20032619055164613425568765078, 2.54773971534484745377811429033, 4.10491695588182596053675766811, 4.62101221991804696586040385955, 5.16844740100516949021650657759, 6.22260749729063194198420768014, 7.21502769034506298934328176616, 8.140560210250584206848825010845, 8.848230347662015276069176611433, 9.919402579247381229650396525367

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