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Properties

Label 2-2175-435.434-c0-0-4
Degree $2$
Conductor $2175$
Sign $0.447 - 0.894i$
Analytic cond. $1.08546$
Root an. cond. $1.04185$
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  + 0.347i·2-s − i·3-s + 0.879·4-s + 0.347·6-s + 1.87i·7-s + 0.652i·8-s − 9-s − 1.53·11-s − 0.879i·12-s + 0.347i·13-s − 0.652·14-s + 0.652·16-s + 1.53i·17-s − 0.347i·18-s + 1.87·21-s − 0.532i·22-s + ⋯
L(s)  = 1  + 0.347i·2-s − i·3-s + 0.879·4-s + 0.347·6-s + 1.87i·7-s + 0.652i·8-s − 9-s − 1.53·11-s − 0.879i·12-s + 0.347i·13-s − 0.652·14-s + 0.652·16-s + 1.53i·17-s − 0.347i·18-s + 1.87·21-s − 0.532i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.08546\)
Root analytic conductor: \(1.04185\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232522514\)
\(L(\frac12)\) \(\approx\) \(1.232522514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
29 \( 1 - T \)
good2 \( 1 - 0.347iT - T^{2} \)
7 \( 1 - 1.87iT - T^{2} \)
11 \( 1 + 1.53T + T^{2} \)
13 \( 1 - 0.347iT - T^{2} \)
17 \( 1 - 1.53iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.87iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.53iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 0.347T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.047179885302402289033225240344, −8.224500300559437847655053487270, −8.063214235273163556096168692161, −6.99113733584762074763094910187, −6.18074493988751036527167172600, −5.74647972366533448638752400979, −5.04957346393575503913048518832, −3.17888835556466171006101325012, −2.38481037981051107665735447477, −1.87237842216213082542263440373, 0.811823675829709997785431308179, 2.63978715530605962633459164257, 3.16310656184559475310269913081, 4.23406977267204944449113408011, 4.91249079498786374679198883511, 5.88289268434447235264471435553, 6.95576065887986242762956142598, 7.56871105815980012134461545197, 8.174871064809424412150866223821, 9.554441044523615279202614159107

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