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Properties

Label 2-1295-7.2-c1-0-20
Degree $2$
Conductor $1295$
Sign $-0.998 + 0.0543i$
Analytic cond. $10.3406$
Root an. cond. $3.21568$
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  + (−0.0342 + 0.0592i)2-s + (1.52 + 2.64i)3-s + (0.997 + 1.72i)4-s + (−0.5 + 0.866i)5-s − 0.208·6-s + (−2.27 − 1.34i)7-s − 0.273·8-s + (−3.14 + 5.45i)9-s + (−0.0342 − 0.0592i)10-s + (0.902 + 1.56i)11-s + (−3.04 + 5.26i)12-s + 1.04·13-s + (0.157 − 0.0890i)14-s − 3.04·15-s + (−1.98 + 3.43i)16-s + (0.286 + 0.495i)17-s + ⋯
L(s)  = 1  + (−0.0241 + 0.0418i)2-s + (0.880 + 1.52i)3-s + (0.498 + 0.863i)4-s + (−0.223 + 0.387i)5-s − 0.0851·6-s + (−0.861 − 0.507i)7-s − 0.0966·8-s + (−1.04 + 1.81i)9-s + (−0.0108 − 0.0187i)10-s + (0.272 + 0.471i)11-s + (−0.877 + 1.52i)12-s + 0.289·13-s + (0.0421 − 0.0238i)14-s − 0.787·15-s + (−0.496 + 0.859i)16-s + (0.0693 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $-0.998 + 0.0543i$
Analytic conductor: \(10.3406\)
Root analytic conductor: \(3.21568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1295} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1295,\ (\ :1/2),\ -0.998 + 0.0543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.890264080\)
\(L(\frac12)\) \(\approx\) \(1.890264080\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.27 + 1.34i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.0342 - 0.0592i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.52 - 2.64i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.902 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 + (-0.286 - 0.495i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.411 - 0.712i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 + (0.735 + 1.27i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 - 2.41T + 41T^{2} \)
43 \( 1 - 4.47T + 43T^{2} \)
47 \( 1 + (0.424 - 0.735i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.836 - 1.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.17 - 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.37 + 2.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 - 5.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.49T + 71T^{2} \)
73 \( 1 + (3.00 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.51 + 7.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.55T + 83T^{2} \)
89 \( 1 + (2.60 - 4.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.36T + 97T^{2} \)
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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

−10.06413885225055227957961855356, −9.223366886993802002586401253001, −8.626124526449867328200444889284, −7.69763956465944964691445570315, −6.99784887644757129050535489763, −5.96325066804804824631895244707, −4.49740006984741227770545677716, −3.88210213274759023055036739512, −3.22868101956403805288419735050, −2.40414224814384087198453863269, 0.68362745409456409304696208972, 1.75133860483721670850198592924, 2.71184777978147620949529800861, 3.61599354194800062268102049622, 5.38626335810959511337543589374, 6.12163724655386657514785298416, 6.82455766193163232075018305451, 7.48587585599952208465723507994, 8.476630785684855097876799532424, 9.132284538368975160426268787707

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