Properties

Label 2-3332-476.243-c0-0-0
Degree $2$
Conductor $3332$
Sign $0.379 - 0.925i$
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  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.09 + 1.25i)5-s + (−0.923 − 0.382i)8-s + (0.608 + 0.793i)9-s + (1.25 − 1.09i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (−1.38 + 0.923i)20-s + (−0.230 − 1.75i)25-s + (−1.12 + 0.860i)26-s + (1.63 + 0.324i)29-s + (−0.793 − 0.608i)32-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.09 + 1.25i)5-s + (−0.923 − 0.382i)8-s + (0.608 + 0.793i)9-s + (1.25 − 1.09i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (−1.38 + 0.923i)20-s + (−0.230 − 1.75i)25-s + (−1.12 + 0.860i)26-s + (1.63 + 0.324i)29-s + (−0.793 − 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7362905831\)
\(L(\frac12)\) \(\approx\) \(0.7362905831\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 \)
17 \( 1 + (-0.793 - 0.608i)T \)
good3 \( 1 + (-0.608 - 0.793i)T^{2} \)
5 \( 1 + (1.09 - 1.25i)T + (-0.130 - 0.991i)T^{2} \)
11 \( 1 + (0.991 + 0.130i)T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + (0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.608 - 0.793i)T^{2} \)
29 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.608 + 0.793i)T^{2} \)
37 \( 1 + (-1.10 + 0.0726i)T + (0.991 - 0.130i)T^{2} \)
41 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.369 + 0.125i)T + (0.793 + 0.608i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.357 - 1.05i)T + (-0.793 + 0.608i)T^{2} \)
79 \( 1 + (-0.608 + 0.793i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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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

−8.670887506489250684401579640540, −8.008294596894279234747075171165, −7.79467216575333618187438625636, −6.82936672075167477825965292241, −6.38237905686025769030136248772, −5.25018032819728443864873653953, −3.91347892450026708685576745645, −3.30778366561517825425808390160, −2.49450625595718552119242791782, −1.15359923355206541403725485313, 0.78485564838477890439651193231, 1.51458560716832772343097917130, 3.10134038671421883305311767163, 4.02095423389818680471157437974, 4.73603477297703203900006386206, 5.82362832064963707846592908577, 6.66601933222541661118361000784, 7.32439364993219009441121442675, 8.133899391967013502234074038055, 8.646280414495032928229046279336

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