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Properties

Label 2-2178-33.32-c1-0-28
Degree $2$
Conductor $2178$
Sign $0.394 + 0.918i$
Analytic cond. $17.3914$
Root an. cond. $4.17030$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.93i·5-s + 8-s − 1.93i·10-s + 0.517i·13-s + 16-s + 5·17-s − 7.72i·19-s − 1.93i·20-s + 2.07i·23-s + 1.26·25-s + 0.517i·26-s − 9.19·29-s + 6.92·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.863i·5-s + 0.353·8-s − 0.610i·10-s + 0.143i·13-s + 0.250·16-s + 1.21·17-s − 1.77i·19-s − 0.431i·20-s + 0.431i·23-s + 0.253·25-s + 0.101i·26-s − 1.70·29-s + 1.24·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2178\)    =    \(2 \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(17.3914\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2178} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2178,\ (\ :1/2),\ 0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.778102611\)
\(L(\frac12)\) \(\approx\) \(2.778102611\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + 1.93iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 - 0.517iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + 7.72iT - 19T^{2} \)
23 \( 1 - 2.07iT - 23T^{2} \)
29 \( 1 + 9.19T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 3.92T + 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 - 5.65iT - 47T^{2} \)
53 \( 1 + 13.9iT - 53T^{2} \)
59 \( 1 + 5.65iT - 59T^{2} \)
61 \( 1 + 0.656iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + 9.76iT - 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−8.946963758917016405094304313466, −8.116051521029897538477355746524, −7.26386357121182202358426287588, −6.55573769948817457491935602936, −5.37390533499086573597140310030, −5.11872008594726770986286789158, −4.08188844781428505550324418319, −3.21689639036868521834749474389, −2.06883190453777404533758059520, −0.803625489307882722692235581016, 1.43236139775505650666772353284, 2.66908167048025799867612159982, 3.44435835286657161859150743897, 4.19949759254518242977440074848, 5.40263634916511925684820893392, 5.95450788946463613523827597741, 6.77350701994811195254719414967, 7.60549017515265236732031542665, 8.144850171725450778801004039333, 9.327695889999472012527145175632

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