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Properties

Label 2-1575-105.62-c0-0-6
Degree $2$
Conductor $1575$
Sign $0.749 + 0.662i$
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  + (1 − i)2-s − i·4-s + (0.707 + 0.707i)7-s + 1.41i·11-s + 1.41·14-s + 16-s + (1.41 + 1.41i)22-s + (−1 − i)23-s + (0.707 − 0.707i)28-s − 1.41·29-s + (1 − i)32-s + (−1.41 − 1.41i)37-s + (1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯
L(s)  = 1  + (1 − i)2-s − i·4-s + (0.707 + 0.707i)7-s + 1.41i·11-s + 1.41·14-s + 16-s + (1.41 + 1.41i)22-s + (−1 − i)23-s + (0.707 − 0.707i)28-s − 1.41·29-s + (1 − i)32-s + (−1.41 − 1.41i)37-s + (1.41 − 1.41i)43-s + 1.41·44-s − 2·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.003804582\)
\(L(\frac12)\) \(\approx\) \(2.003804582\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1 + i)T - iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.734607599617131855971131472693, −8.868116735997220002653907211757, −7.908802725292050503752832779602, −7.14548019243166104953283662636, −5.86943550999571355956861441954, −5.16386153263934164796902579443, −4.40736471351205059243657853202, −3.64624729068041528649182521423, −2.24318965791129393154419731591, −1.92967122841025995921833626643, 1.42859601942606664496582095775, 3.25025122499728783537351182356, 3.97280998433675398358814642686, 4.85393285231837693529102005690, 5.70434431063958509331185228818, 6.24250962183797518004255215219, 7.30946607731552893033521420475, 7.81828459066011525662042683404, 8.577980179260584985819491918164, 9.678437011252515424707158164037

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