Properties

Degree $2$
Conductor $1575$
Sign $0.342 - 0.939i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + i·17-s + (−0.5 − 0.866i)21-s i·27-s + 0.999i·28-s + (1 + 1.73i)29-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯
L(s)  = 1  + i·3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + i·17-s + (−0.5 − 0.866i)21-s i·27-s + 0.999i·28-s + (1 + 1.73i)29-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.342 - 0.939i$
Motivic weight: \(0\)
Character: $\chi_{1575} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120891360\)
\(L(\frac12)\) \(\approx\) \(1.120891360\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.882945718684564765977394015568, −9.061509463514713855699468696296, −8.612159120527712286954596167642, −7.10948742765754510011403489520, −6.36043914413746361929217544352, −5.77847644038558579992083573489, −4.82148606476732153881546481254, −3.88473803549106906982580107473, −2.89592097041120463039240858315, −1.65675978560010566305981127842, 0.933100740722115110091882544548, 2.57117304024433075536279854819, 3.22163759071610718594362779710, 4.14384563040902869233923008956, 5.79071754921210242733575081470, 6.37228678742432010156662628566, 7.04617732380829806092168981110, 7.81416484697865207993431669241, 8.476150256186513029777450972853, 9.234390226051045123538238514807

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