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Properties

Label 2-2175-435.434-c0-0-1
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  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s + 1.53i·7-s + 2.87i·8-s − 9-s + 0.347·11-s + 2.53i·12-s + 1.87i·13-s + 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s + 1.53·21-s − 0.652i·22-s + ⋯
L(s)  = 1  − 1.87i·2-s − i·3-s − 2.53·4-s − 1.87·6-s + 1.53i·7-s + 2.87i·8-s − 9-s + 0.347·11-s + 2.53i·12-s + 1.87i·13-s + 2.87·14-s + 2.87·16-s + 0.347i·17-s + 1.87i·18-s + 1.53·21-s − 0.652i·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

\(L(\frac{1}{2})\) \(\approx\) \(0.7278357022\)
\(L(\frac12)\) \(\approx\) \(0.7278357022\)
\(L(1)\) 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 \( 1 + 1.87iT - T^{2} \)
7 \( 1 - 1.53iT - T^{2} \)
11 \( 1 - 0.347T + T^{2} \)
13 \( 1 - 1.87iT - T^{2} \)
17 \( 1 - 0.347iT - T^{2} \)
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 \( 1 - 1.53iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 0.347iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.87T + T^{2} \)
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

−9.016540890858214709683780143061, −8.893089851778663291301470149533, −7.917044891572405954292589723795, −6.67414426958010706697138107588, −5.83498347444408225519383965056, −4.93310556136923982843168318248, −3.92038011284746429395361893673, −2.89900020872895979362590543461, −2.06142696728767120350601762134, −1.56395519216789662685669275993, 0.52974695487465657582815957331, 3.37782493602904219011034578546, 3.92227413147199346103681984728, 4.87521113036297526372731130831, 5.40752592744712940644653307963, 6.25766578121597719620771734941, 7.15061526408470594627887254430, 7.75605135970267273892056290652, 8.394532890727458350181657618729, 9.195026941475063955003507763164

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