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Properties

Label 2-1575-105.62-c0-0-2
Degree $2$
Conductor $1575$
Sign $-0.465 - 0.884i$
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  + (−1.36 + 1.36i)2-s − 2.73i·4-s + (0.707 + 0.707i)7-s + (2.36 + 2.36i)8-s + 0.517i·11-s − 1.93·14-s − 3.73·16-s + (−0.707 − 0.707i)22-s + (−0.366 − 0.366i)23-s + (1.93 − 1.93i)28-s + 1.93·29-s + (2.73 − 2.73i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s + 1.41·44-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s − 2.73i·4-s + (0.707 + 0.707i)7-s + (2.36 + 2.36i)8-s + 0.517i·11-s − 1.93·14-s − 3.73·16-s + (−0.707 − 0.707i)22-s + (−0.366 − 0.366i)23-s + (1.93 − 1.93i)28-s + 1.93·29-s + (2.73 − 2.73i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s + 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.465 - 0.884i$
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: \((2,\ 1575,\ (\ :0),\ -0.465 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5888021875\)
\(L(\frac12)\) \(\approx\) \(0.5888021875\)
\(L(1)\) 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 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
29 \( 1 - 1.93T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 - 1.93iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + iT - T^{2} \)
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

−9.784530900395072950390031411575, −8.759836154054357132509957237561, −8.324622837995334514119927674383, −7.68598354963965700613995410414, −6.72772627390866816784191880329, −6.16256035830108385179713263002, −5.17660578097449998253715183679, −4.56947592493747470899075214247, −2.45945762114848560825649365176, −1.29994934193556921827966284901, 0.861828767446607314889184561646, 1.93768012058986991320882912297, 3.04164116748702008826977240626, 3.92670879058723171195278808026, 4.83653178894224350905498506892, 6.44154199985125908027307548908, 7.44769449073133678428575791088, 8.055054408011781176650335139948, 8.649642846639026022931520492213, 9.496002656742637337611491673342

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