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Properties

Label 2-3332-476.227-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.995 - 0.0902i$
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  + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.128 − 1.95i)5-s + (0.382 + 0.923i)8-s + (0.793 + 0.608i)9-s + (−1.95 + 0.128i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (0.499 − 0.866i)18-s + (0.382 + 1.92i)20-s + (−2.82 − 0.371i)25-s + (−0.860 + 1.12i)26-s + (0.324 + 0.216i)29-s + (−0.608 − 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.128 − 1.95i)5-s + (0.382 + 0.923i)8-s + (0.793 + 0.608i)9-s + (−1.95 + 0.128i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (0.499 − 0.866i)18-s + (0.382 + 1.92i)20-s + (−2.82 − 0.371i)25-s + (−0.860 + 1.12i)26-s + (0.324 + 0.216i)29-s + (−0.608 − 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.995 - 0.0902i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.995 - 0.0902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9779899887\)
\(L(\frac12)\) \(\approx\) \(0.9779899887\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 + 0.991i)T \)
7 \( 1 \)
17 \( 1 + (-0.793 + 0.608i)T \)
good3 \( 1 + (-0.793 - 0.608i)T^{2} \)
5 \( 1 + (-0.128 + 1.95i)T + (-0.991 - 0.130i)T^{2} \)
11 \( 1 + (-0.130 - 0.991i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-0.965 + 0.258i)T^{2} \)
23 \( 1 + (0.793 - 0.608i)T^{2} \)
29 \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.793 + 0.608i)T^{2} \)
37 \( 1 + (1.29 + 1.47i)T + (-0.130 + 0.991i)T^{2} \)
41 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.735 - 1.49i)T + (-0.608 - 0.793i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.349 + 0.172i)T + (0.608 - 0.793i)T^{2} \)
79 \( 1 + (-0.793 + 0.608i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \)
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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

−8.699346056842022560304372827232, −7.77515247119372625154644617391, −7.42998934311367461380367571388, −5.56173693103447641639229350512, −5.24607254652658368152613287368, −4.56077836255676523856813432319, −3.81552564913571238786868032972, −2.54379638512723395803426812411, −1.59581362748045114083042688336, −0.64194838965902721018258893059, 1.70528668479399143179754971210, 2.98722116451874023750719616014, 3.79888745914674970703245650997, 4.59811811871307873189479492823, 5.74123549773404678811008339413, 6.50447167846533544883961381713, 6.87451956849546724134137307480, 7.45196364180789625350468758166, 8.161227759571782319875449057223, 9.311964217085988540160790094222

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