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Properties

Label 2-1078-7.2-c1-0-14
Degree $2$
Conductor $1078$
Sign $0.947 + 0.318i$
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 + (−0.499 − 0.866i)4-s + (−1.41 + 2.44i)5-s − 0.999·8-s + (1.5 − 2.59i)9-s + (1.41 + 2.44i)10-s + (−0.5 − 0.866i)11-s + 4.24·13-s + (−0.5 + 0.866i)16-s + (1.41 + 2.44i)17-s + (−1.5 − 2.59i)18-s + (−0.707 + 1.22i)19-s + 2.82·20-s − 0.999·22-s + (−3 + 5.19i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.632 + 1.09i)5-s − 0.353·8-s + (0.5 − 0.866i)9-s + (0.447 + 0.774i)10-s + (−0.150 − 0.261i)11-s + 1.17·13-s + (−0.125 + 0.216i)16-s + (0.342 + 0.594i)17-s + (−0.353 − 0.612i)18-s + (−0.162 + 0.280i)19-s + 0.632·20-s − 0.213·22-s + (−0.625 + 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.947 + 0.318i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800583764\)
\(L(\frac12)\) \(\approx\) \(1.800583764\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.41 - 2.44i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.707 - 1.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3.53 + 6.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.07 - 12.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.12 + 3.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (4.24 + 7.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.07T + 83T^{2} \)
89 \( 1 + (9.19 - 15.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.41T + 97T^{2} \)
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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

−10.11103428658456344443418473842, −9.148987998897783123823620290008, −8.195567001991814992981760763733, −7.27320174404520675152109415142, −6.35061119445205054501597976959, −5.69911591630156943154809195778, −4.04187162951966395013939553579, −3.72167320801623320667003979329, −2.67589720877156500588349331755, −1.11463016822784494115451560899, 0.962355235283856252285196608702, 2.68856976519021153669514494598, 4.29349731444503813491053326508, 4.48848658430210314002179695683, 5.57930629439861594273823528090, 6.52373273568344765018038788274, 7.58647727550995795652795793871, 8.198091758579881158835442893114, 8.777160794017142484405888657008, 9.803199950905744246532560810729

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