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Properties

Label 2-3332-3332.611-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.138 + 0.990i$
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  + (0.0747 − 0.997i)2-s + (1.82 − 0.563i)3-s + (−0.988 − 0.149i)4-s + (−0.425 − 1.86i)6-s + (−0.222 + 0.974i)7-s + (−0.222 + 0.974i)8-s + (2.19 − 1.49i)9-s + (0.123 + 0.0841i)11-s + (−1.88 + 0.284i)12-s + (−0.134 + 0.0648i)13-s + (0.955 + 0.294i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−1.32 − 2.29i)18-s + (0.142 + 1.90i)21-s + (0.0931 − 0.116i)22-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (1.82 − 0.563i)3-s + (−0.988 − 0.149i)4-s + (−0.425 − 1.86i)6-s + (−0.222 + 0.974i)7-s + (−0.222 + 0.974i)8-s + (2.19 − 1.49i)9-s + (0.123 + 0.0841i)11-s + (−1.88 + 0.284i)12-s + (−0.134 + 0.0648i)13-s + (0.955 + 0.294i)14-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−1.32 − 2.29i)18-s + (0.142 + 1.90i)21-s + (0.0931 − 0.116i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.238794720\)
\(L(\frac12)\) \(\approx\) \(2.238794720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 + 0.997i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
good3 \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 + 0.294i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.715766208756300307410375148581, −8.219469203684970407000506231368, −7.36744395939451979789925610647, −6.51831399897848205726747885479, −5.38193791748714577164553993903, −4.37508788872804051760674880803, −3.55779940686256011106135857630, −2.68801468291710407811873272939, −2.37980056555390516334230007232, −1.27134875628648076399096018637, 1.54217661353830773718073996063, 2.91577430461280319023609271984, 3.87409269413882433790114345419, 4.01861442956785365119411768031, 5.05671941706251804392176200881, 6.14609752127566784911163685219, 7.07959256962220705771968326601, 7.75495573670417432165360348978, 8.074213887203973433315587380315, 8.891608261987623576608509042534

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