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Properties

Label 2-1575-15.2-c1-0-7
Degree $2$
Conductor $1575$
Sign $-0.749 - 0.662i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 1.36i)2-s − 1.73i·4-s + (−0.707 − 0.707i)7-s + (−0.366 − 0.366i)8-s + 2.44i·11-s + (1.03 − 1.03i)13-s + 1.93·14-s + 4.46·16-s + (−2 + 2i)17-s − 2i·19-s + (−3.34 − 3.34i)22-s + (−0.464 − 0.464i)23-s + 2.82i·26-s + (−1.22 + 1.22i)28-s + 3.48·29-s + ⋯
L(s)  = 1  + (−0.965 + 0.965i)2-s − 0.866i·4-s + (−0.267 − 0.267i)7-s + (−0.129 − 0.129i)8-s + 0.738i·11-s + (0.287 − 0.287i)13-s + 0.516·14-s + 1.11·16-s + (−0.485 + 0.485i)17-s − 0.458i·19-s + (−0.713 − 0.713i)22-s + (−0.0967 − 0.0967i)23-s + 0.554i·26-s + (−0.231 + 0.231i)28-s + 0.647·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.749 - 0.662i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7135667658\)
\(L(\frac12)\) \(\approx\) \(0.7135667658\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 1.36i)T - 2iT^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (-1.03 + 1.03i)T - 13iT^{2} \)
17 \( 1 + (2 - 2i)T - 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (0.464 + 0.464i)T + 23iT^{2} \)
29 \( 1 - 3.48T + 29T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 + (5.27 + 5.27i)T + 37iT^{2} \)
41 \( 1 - 7.72iT - 41T^{2} \)
43 \( 1 + (2.82 - 2.82i)T - 43iT^{2} \)
47 \( 1 + (0.535 - 0.535i)T - 47iT^{2} \)
53 \( 1 + (-5.73 - 5.73i)T + 53iT^{2} \)
59 \( 1 + 6.96T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + (-3.48 - 3.48i)T + 67iT^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (3.86 - 3.86i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + (-11.4 - 11.4i)T + 83iT^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-13.6 - 13.6i)T + 97iT^{2} \)
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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

−9.664981322944959296950945030626, −8.717058140904435426711132033341, −8.211014324666885728717634669795, −7.34565911500145691754957796853, −6.67537024320472579207208119397, −6.07152870269218720991530504681, −4.91772865366756717214385175742, −3.89900989319116798220390656547, −2.66081313543374789112266804694, −1.06192791760985404331031506463, 0.46498888877697089979319011004, 1.74591354667854365758835895559, 2.79530257595619480163282864658, 3.60286795025651801031490287422, 4.91471783408045882427109327624, 5.94770373140401351743503367402, 6.72714201211199563624478864609, 7.919794219918216827390104116225, 8.644300416948005651639036532557, 9.077582064198251294696668881047

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