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Properties

Label 2-5376-8.5-c1-0-67
Degree $2$
Conductor $5376$
Sign $-0.707 + 0.707i$
Analytic cond. $42.9275$
Root an. cond. $6.55191$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 2i·5-s − 7-s − 9-s + 4i·11-s + 6i·13-s + 2·15-s − 2·17-s + 4i·19-s − i·21-s − 4·23-s + 25-s − i·27-s + 2i·29-s − 8·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.894i·5-s − 0.377·7-s − 0.333·9-s + 1.20i·11-s + 1.66i·13-s + 0.516·15-s − 0.485·17-s + 0.917i·19-s − 0.218i·21-s − 0.834·23-s + 0.200·25-s − 0.192i·27-s + 0.371i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5376\)    =    \(2^{8} \cdot 3 \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(42.9275\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5376} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 5376,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
7 \( 1 + T \)
good5 \( 1 + 2iT - 5T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.980429295829972763339150929376, −7.12848448210342444478720069643, −6.55197057222954709846150037150, −5.54797915635949854005537895631, −4.96385823607857016413063572385, −4.01473040705127168586474644964, −3.88383371842442626249107445350, −2.25480581246777942234856344863, −1.67371036811037920712114210785, 0, 1.07402376152324713741471343362, 2.54705289132506810292487891514, 2.98805194554284973854058449427, 3.71516227340279091913054824687, 4.95201737198047055716805399331, 5.83618228497125908807436319511, 6.26426247803484509552425333969, 7.00005604742938908064987684111, 7.72213915376762324295688574673

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