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Properties

Label 2-3328-104.43-c0-0-1
Degree $2$
Conductor $3328$
Sign $0.962 - 0.271i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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  + 1.73·5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)17-s + 1.99·25-s + (−0.866 − 0.5i)29-s + (−0.866 + 1.5i)37-s + (−1.5 − 0.866i)41-s + (0.866 + 1.49i)45-s + (0.5 − 0.866i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (1.49 + 0.866i)65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + 1.73·5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.5 − 0.866i)17-s + 1.99·25-s + (−0.866 − 0.5i)29-s + (−0.866 + 1.5i)37-s + (−1.5 − 0.866i)41-s + (0.866 + 1.49i)45-s + (0.5 − 0.866i)49-s + i·53-s + (−0.866 + 0.5i)61-s + (1.49 + 0.866i)65-s − 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.831733122\)
\(L(\frac12)\) \(\approx\) \(1.831733122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - 1.73T + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.994368965853480632322753157548, −8.231076217736034808264363015376, −7.13475582144705886815367481054, −6.61683972885916007779331225338, −5.77028921888019134293472272984, −5.15489637245666936994068065030, −4.37668009718991208875640079204, −3.12214140520739291466656349057, −2.08452866486590313872230334068, −1.53952168311849149372087776395, 1.31449899475662082895194748476, 2.02642201263275623291072426092, 3.21340357993861212967020441875, 4.03178029823154570464843387564, 5.20684339759680131101763694665, 5.82405403479312732800529576482, 6.44543488964390658280874747200, 7.02291273677693512981666705811, 8.203164438757058843408125244945, 8.990725861206523963717906930237

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