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Properties

Label 2-2175-5.4-c1-0-24
Degree $2$
Conductor $2175$
Sign $-0.894 - 0.447i$
Analytic cond. $17.3674$
Root an. cond. $4.16742$
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  + 1.79i·2-s − i·3-s − 1.20·4-s + 1.79·6-s + i·7-s + 1.41i·8-s − 9-s + 5·11-s + 1.20i·12-s + 4.58i·13-s − 1.79·14-s − 4.95·16-s − 3i·17-s − 1.79i·18-s − 3.58·19-s + ⋯
L(s)  = 1  + 1.26i·2-s − 0.577i·3-s − 0.604·4-s + 0.731·6-s + 0.377i·7-s + 0.501i·8-s − 0.333·9-s + 1.50·11-s + 0.348i·12-s + 1.27i·13-s − 0.478·14-s − 1.23·16-s − 0.727i·17-s − 0.422i·18-s − 0.821·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2175\)    =    \(3 \cdot 5^{2} \cdot 29\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(17.3674\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2175} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2175,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658103881\)
\(L(\frac12)\) \(\approx\) \(1.658103881\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
29 \( 1 + T \)
good2 \( 1 - 1.79iT - 2T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 3.58T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 9.16T + 41T^{2} \)
43 \( 1 - 9.58iT - 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 + 0.417iT - 53T^{2} \)
59 \( 1 - 7.58T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 4.16iT - 67T^{2} \)
71 \( 1 + 9.58T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 7.58T + 79T^{2} \)
83 \( 1 - 11.5iT - 83T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−9.065106364759053554227844637744, −8.552990298507929512903568142319, −7.65522005202914575443512844989, −6.85675038928827682182166198132, −6.52025793371046685441787730877, −5.78772599731370243859779649171, −4.79646561414338014838608766048, −3.92994855302328972088214128811, −2.49355935340586187300617968016, −1.46564327400613206822686900070, 0.58698656628805986390457317485, 1.76332515639060866721098504392, 2.87039569639614589938371165287, 3.82146709273995111677361717207, 4.16357382058152508737311638387, 5.35506857290199480795398613436, 6.42693224397819184236512680534, 7.04388804678459295085891399283, 8.476225949962167607138452332631, 8.753101601478421218480050782184

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