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Properties

Label 2-2175-5.4-c1-0-62
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  + 1.46i·2-s + i·3-s − 0.139·4-s − 1.46·6-s − 1.32i·7-s + 2.72i·8-s − 9-s − 5.32·11-s − 0.139i·12-s − 2.39i·13-s + 1.93·14-s − 4.25·16-s − 5.04i·17-s − 1.46i·18-s − 0.925·19-s + ⋯
L(s)  = 1  + 1.03i·2-s + 0.577i·3-s − 0.0695·4-s − 0.597·6-s − 0.500i·7-s + 0.962i·8-s − 0.333·9-s − 1.60·11-s − 0.0401i·12-s − 0.665i·13-s + 0.517·14-s − 1.06·16-s − 1.22i·17-s − 0.344i·18-s − 0.212·19-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

\(L(1)\) \(\approx\) \(0.8300362174\)
\(L(\frac12)\) \(\approx\) \(0.8300362174\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.46iT - 2T^{2} \)
7 \( 1 + 1.32iT - 7T^{2} \)
11 \( 1 + 5.32T + 11T^{2} \)
13 \( 1 + 2.39iT - 13T^{2} \)
17 \( 1 + 5.04iT - 17T^{2} \)
19 \( 1 + 0.925T + 19T^{2} \)
23 \( 1 + 5.72iT - 23T^{2} \)
31 \( 1 - 5.72T + 31T^{2} \)
37 \( 1 - 1.07iT - 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 9.29iT - 43T^{2} \)
47 \( 1 - 3.97iT - 47T^{2} \)
53 \( 1 + 2.12iT - 53T^{2} \)
59 \( 1 - 1.35T + 59T^{2} \)
61 \( 1 - 2.92T + 61T^{2} \)
67 \( 1 + 7.97iT - 67T^{2} \)
71 \( 1 + 4.12T + 71T^{2} \)
73 \( 1 - 1.07iT - 73T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 - 4.79iT - 83T^{2} \)
89 \( 1 - 0.547T + 89T^{2} \)
97 \( 1 + 5.57iT - 97T^{2} \)
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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

−8.631131851809854901136430101024, −8.215455319851633369893916979320, −7.39118418651607452502564061080, −6.80452674875797708079547378078, −5.77361658007636918230834627859, −5.13296826981725685446076491890, −4.53120394212220484651355667415, −3.11197416585201466789715223303, −2.38448154626074029752869466271, −0.26879981245029802055775090508, 1.40733691674606719290876071417, 2.24196874558558341165232043845, 2.98741904847059349632911037729, 3.97562649208654918004964523541, 5.12785789919476738080718129908, 5.99920056319391399024084540830, 6.78686518101722973615468500502, 7.67453919122002576909040422829, 8.357715211075908363970591381340, 9.221922859734647951064018647777

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