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Properties

Label 2-1596-1596.83-c0-0-2
Degree $2$
Conductor $1596$
Sign $0.977 + 0.211i$
Analytic cond. $0.796507$
Root an. cond. $0.892472$
Motivic weight $0$
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 11-s + 0.999·12-s + (0.5 + 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + 11-s + 0.999·12-s + (0.5 + 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.796507\)
Root analytic conductor: \(0.892472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1596} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1596,\ (\ :0),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156705845\)
\(L(\frac12)\) \(\approx\) \(1.156705845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 - T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.060283980223719389266635941910, −8.383838797923700874884287492424, −8.103225046207645987830121605288, −6.91170414691845306254359946523, −6.58000993713222950167333036663, −5.40792953057127450588370751030, −4.74280336816020163240315817064, −4.13596127480650960361179811252, −2.50997736454167739372411746163, −0.972733321512063517460296557456, 1.44260092666158496693050292945, 2.92932280342745345758789338625, 3.74399984198466322174026167947, 4.50047523684362175703236578705, 5.19377351736445941863355114817, 6.27053216134840864865462649175, 6.94300837519052736787524490748, 8.329647868614934950875684090772, 9.067886951137611438718453957655, 9.841925487486072338712976126604

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