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Properties

Label 2-3332-68.67-c0-0-18
Degree $2$
Conductor $3332$
Sign $-0.587 + 0.809i$
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  + (0.309 − 0.951i)2-s + 1.90·3-s + (−0.809 − 0.587i)4-s − 1.61i·5-s + (0.587 − 1.80i)6-s + (−0.809 + 0.587i)8-s + 2.61·9-s + (−1.53 − 0.500i)10-s + (−1.53 − 1.11i)12-s − 3.07i·15-s + (0.309 + 0.951i)16-s + i·17-s + (0.809 − 2.48i)18-s + (−0.951 + 1.30i)20-s + (−1.53 + 1.11i)24-s − 1.61·25-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + 1.90·3-s + (−0.809 − 0.587i)4-s − 1.61i·5-s + (0.587 − 1.80i)6-s + (−0.809 + 0.587i)8-s + 2.61·9-s + (−1.53 − 0.500i)10-s + (−1.53 − 1.11i)12-s − 3.07i·15-s + (0.309 + 0.951i)16-s + i·17-s + (0.809 − 2.48i)18-s + (−0.951 + 1.30i)20-s + (−1.53 + 1.11i)24-s − 1.61·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.587 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.614609408\)
\(L(\frac12)\) \(\approx\) \(2.614609408\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 \)
17 \( 1 - iT \)
good3 \( 1 - 1.90T + T^{2} \)
5 \( 1 + 1.61iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.17T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 0.618iT - T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618iT - T^{2} \)
67 \( 1 - 1.17iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.61iT - T^{2} \)
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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

−8.659809293355819376971711797389, −8.232122495829021704528762105539, −7.49591625831577154996850893141, −6.12715162125488854988308582382, −5.07248586620730395987367753258, −4.34913444195663308746030702772, −3.80571261912055312701703641187, −2.93468934792038077756834811203, −1.87578178707204606551755902552, −1.31954123019773204741349016328, 2.08121405490845365240587063667, 2.97023712705228218343347982686, 3.42849716697419220056990482517, 4.18602802320557779062985723159, 5.26512399048261883460316734087, 6.51859572088121423907463438326, 6.96557342566697063677028223628, 7.60883454677371859332484502176, 8.032669590253734839710745077976, 9.019589578505062561339304313653

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