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Properties

Label 2-1078-7.4-c1-0-27
Degree $2$
Conductor $1078$
Sign $-0.900 - 0.435i$
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 + (−0.707 + 1.22i)3-s + (−0.499 + 0.866i)4-s + (−2.12 − 3.67i)5-s + 1.41·6-s + 0.999·8-s + (0.500 + 0.866i)9-s + (−2.12 + 3.67i)10-s + (0.5 − 0.866i)11-s + (−0.707 − 1.22i)12-s + 6·15-s + (−0.5 − 0.866i)16-s + (2.82 − 4.89i)17-s + (0.499 − 0.866i)18-s + 4.24·20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.408 + 0.707i)3-s + (−0.249 + 0.433i)4-s + (−0.948 − 1.64i)5-s + 0.577·6-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.670 + 1.16i)10-s + (0.150 − 0.261i)11-s + (−0.204 − 0.353i)12-s + 1.54·15-s + (−0.125 − 0.216i)16-s + (0.685 − 1.18i)17-s + (0.117 − 0.204i)18-s + 0.948·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2256999079\)
\(L(\frac12)\) \(\approx\) \(0.2256999079\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.12 + 3.67i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2.82 + 4.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-0.707 + 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (2.12 + 3.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.41 - 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (4.24 - 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−9.622932264419643056374448186191, −8.496412942317778106043445332457, −8.218281649469284335380126380884, −7.14514558720728711668102465556, −5.58745505324211400281561294195, −4.70911807006801719758268150721, −4.34360222680365389620298447198, −3.16051524306858387648923877109, −1.39861236430928194746645600693, −0.12877449719525281497634525229, 1.70242775264785880087716475833, 3.30474698474347804759164360260, 4.08065735938503601543205085087, 5.66249895702367805770359502572, 6.47817717615397359755867292861, 6.96075729797655424594791704208, 7.71643245441955268406703975218, 8.265543480491468488375416143855, 9.697047007462055140865701942212, 10.27808716957043006502343809939

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