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Properties

Label 2-3332-476.27-c0-0-0
Degree $2$
Conductor $3332$
Sign $0.977 + 0.210i$
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  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.324 + 1.63i)5-s + (−0.923 + 0.382i)8-s + (−0.382 − 0.923i)9-s + (1.63 + 0.324i)10-s + (1 + i)13-s + i·16-s + (0.382 − 0.923i)17-s − 18-s + (0.923 − 1.38i)20-s + (−1.63 + 0.675i)25-s + (1.30 − 0.541i)26-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.324 + 1.63i)5-s + (−0.923 + 0.382i)8-s + (−0.382 − 0.923i)9-s + (1.63 + 0.324i)10-s + (1 + i)13-s + i·16-s + (0.382 − 0.923i)17-s − 18-s + (0.923 − 1.38i)20-s + (−1.63 + 0.675i)25-s + (1.30 − 0.541i)26-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.977 + 0.210i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.977 + 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470249688\)
\(L(\frac12)\) \(\approx\) \(1.470249688\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 \)
17 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.382 - 0.923i)T^{2} \)
37 \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
79 \( 1 + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
97 \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \)
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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

−9.057885240598669891034305516719, −8.199062637961145920302320016540, −6.95030343199459181379839793833, −6.47111315093785610595511261729, −5.88676724854155488514789493578, −4.81712714401796877880393027945, −3.74525691457417818661471107468, −3.18872286981222964428811767547, −2.51016142409128919257421190712, −1.30295903260466402709227821052, 0.896810696893167347814059235863, 2.32067262049788211150446961591, 3.68834038120134269828626261613, 4.36937650888599669019209765919, 5.26735305798908241743546707124, 5.68707808114066670785134308259, 6.25757109599282849693322443499, 7.74932354161456704699622248369, 7.915274593011520650486132230098, 8.744750644996974866276943458394

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