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Properties

Label 2-1575-315.167-c0-0-0
Degree $2$
Conductor $1575$
Sign $-0.993 - 0.116i$
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 + (−0.965 − 0.258i)7-s + 1.00i·9-s + (−1.5 + 0.866i)11-s + (−0.258 − 0.965i)12-s + (−0.965 + 0.258i)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 + (−1.67 − 0.448i)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 + (−0.965 − 0.258i)7-s + 1.00i·9-s + (−1.5 + 0.866i)11-s + (−0.258 − 0.965i)12-s + (−0.965 + 0.258i)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 + (−1.67 − 0.448i)33-s + (0.500 − 0.866i)36-s + (−0.866 − 0.500i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.993 - 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2372045039\)
\(L(\frac12)\) \(\approx\) \(0.2372045039\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.809361246409945694479507358519, −9.439855535766803795201982089733, −8.653392463316432335104776279146, −7.64891626509868418568414868687, −6.99054528975964950582512585834, −5.67421876755225563451910948042, −4.72486473381845201007397312986, −4.44190907073575010081299571254, −3.07280553664475052559952547976, −2.28340375124687014419468278309, 0.15858443668964376929472335757, 2.32056093655792188149513431544, 3.07908653980604714570524055063, 3.91039758810007255513451914938, 5.12177037302255399154341578641, 6.03480380939218839394812157106, 6.97601072124518845298666851153, 7.84073577788278312389681699407, 8.443922129728761147244534747804, 8.998136663712683732992566079774

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