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Properties

Label 2-1575-315.272-c0-0-1
Degree $2$
Conductor $1575$
Sign $-0.835 + 0.548i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (−0.707 − 0.707i)3-s + (0.866 − 0.5i)4-s + (−0.258 − 0.965i)7-s + 1.00i·9-s + (−1.5 − 0.866i)11-s + (−0.965 − 0.258i)12-s + (−0.258 + 0.965i)13-s + (0.499 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (−0.500 + 0.866i)21-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)28-s + (0.448 + 1.67i)33-s + (0.500 + 0.866i)36-s + (0.866 − 0.500i)39-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.866 − 0.5i)4-s + (−0.258 − 0.965i)7-s + 1.00i·9-s + (−1.5 − 0.866i)11-s + (−0.965 − 0.258i)12-s + (−0.258 + 0.965i)13-s + (0.499 − 0.866i)16-s + (−1.22 − 1.22i)17-s + (−0.500 + 0.866i)21-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)28-s + (0.448 + 1.67i)33-s + (0.500 + 0.866i)36-s + (0.866 − 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.835 + 0.548i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1532, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.835 + 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7355089791\)
\(L(\frac12)\) \(\approx\) \(0.7355089791\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.258 + 0.965i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.461248397743079137154518066558, −8.263559495761613233243516209317, −7.29348605415994406533590178972, −7.01584084086433003357632594796, −6.15352521743378434471309701672, −5.33252881785656079134601986949, −4.49527011179553502650979801756, −2.90261994390072871772686459368, −2.04931120395844944293695691713, −0.57229582545154634218202707623, 2.17832147218991758898316387840, 2.93477532665304763431442497267, 4.09496224878776000375739147809, 5.14803608813309599254360333013, 5.82867231556417429181087662380, 6.58048590084299269178958579765, 7.54622378776924375524066587330, 8.333912624995482941874246238294, 9.183065093134283808475417113336, 10.33005150656464483164203033463

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