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Properties

Label 2-2175-435.434-c0-0-11
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

\(L(\frac{1}{2})\) \(\approx\) \(1.524096951\)
\(L(\frac12)\) \(\approx\) \(1.524096951\)
\(L(1)\) 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

−8.818733295852830520035133972650, −8.061166441547019587320096293564, −7.41172586148492948770795137387, −6.76796647143613562922316230230, −6.40670099368601936077330933420, −5.44997141458835471154252212047, −3.99792092604343512027324767418, −3.43410170924670516091633517555, −1.88957024790543485481415868619, −1.17871366093912010891378725131, 1.78495838208498490317553901161, 2.74694215368942342595023696109, 3.41521212120751373256652689672, 4.54052864524554951160963920347, 5.47432548594210259895632437752, 6.14484266755393571404332150041, 6.86403047372750676836152242917, 8.017465875846241912170344301256, 9.136425061981662775916504624937, 9.219538216138578897846526317385

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