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Properties

Label 2-3332-68.19-c0-0-0
Degree $2$
Conductor $3332$
Sign $0.739 - 0.673i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.292 + 0.707i)5-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.292 − 0.707i)10-s + 2i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 1.00·18-s + (−0.707 + 0.292i)20-s + (0.292 − 0.292i)25-s + (1.41 − 1.41i)26-s + (0.707 + 1.70i)29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.292 + 0.707i)5-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)9-s + (0.292 − 0.707i)10-s + 2i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 1.00·18-s + (−0.707 + 0.292i)20-s + (0.292 − 0.292i)25-s + (1.41 − 1.41i)26-s + (0.707 + 1.70i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8862506743\)
\(L(\frac12)\) \(\approx\) \(0.8862506743\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 \)
17 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - 2iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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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

−8.905747559005528762401293988171, −8.560957037447836926655235083050, −7.12166805901827682209405060849, −6.94693701199420381132638685521, −6.28704985212214390620944585903, −4.76587169000204867696532546397, −4.02956970599213676033029826720, −3.28716366349171627962390798530, −2.15841504260812958125026092373, −1.46124516365071390515862013025, 0.70358521393780651185275790086, 1.84739879112385287718000702336, 2.95616701517561944424574615945, 4.51237447014751953929728692299, 5.00870124223874830053303421204, 5.73311025379446444825846240177, 6.51769060372961369775851950822, 7.50828873845244617560292049602, 7.921759785439069081325943956238, 8.589208867553381184203361238168

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