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Properties

Label 2-1575-105.59-c1-0-29
Degree $2$
Conductor $1575$
Sign $0.671 - 0.741i$
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  + (0.489 + 0.848i)2-s + (0.520 − 0.900i)4-s + (2.33 + 1.24i)7-s + 2.97·8-s + (4.23 + 2.44i)11-s + 1.19·13-s + (0.0850 + 2.59i)14-s + (0.418 + 0.725i)16-s + (−1.25 − 0.725i)17-s + (−6.58 + 3.80i)19-s + 4.79i·22-s + (3.13 + 5.42i)23-s + (0.586 + 1.01i)26-s + (2.33 − 1.45i)28-s − 0.729i·29-s + ⋯
L(s)  = 1  + (0.346 + 0.599i)2-s + (0.260 − 0.450i)4-s + (0.881 + 0.471i)7-s + 1.05·8-s + (1.27 + 0.737i)11-s + 0.331·13-s + (0.0227 + 0.692i)14-s + (0.104 + 0.181i)16-s + (−0.304 − 0.175i)17-s + (−1.51 + 0.872i)19-s + 1.02i·22-s + (0.653 + 1.13i)23-s + (0.114 + 0.199i)26-s + (0.441 − 0.274i)28-s − 0.135i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.671 - 0.741i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.671 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.788381661\)
\(L(\frac12)\) \(\approx\) \(2.788381661\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.33 - 1.24i)T \)
good2 \( 1 + (-0.489 - 0.848i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-4.23 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 + (1.25 + 0.725i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.58 - 3.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.13 - 5.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.729iT - 29T^{2} \)
31 \( 1 + (8.44 + 4.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.20 + 3.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.08T + 41T^{2} \)
43 \( 1 + 5.20iT - 43T^{2} \)
47 \( 1 + (-7.19 + 4.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.39 - 5.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.08 - 5.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.52 - 2.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.42 + 4.28i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (2.35 - 4.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.81 + 8.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 + (-0.887 - 1.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.2T + 97T^{2} \)
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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

−9.330080762672414816552939904135, −8.812157259873603944714478109162, −7.63398866871665621335095291505, −7.17378891078866633263582496324, −6.06549223148791119333421975008, −5.69847859647044471906892521272, −4.51378039559697532948392582150, −3.99889623622861292245599565057, −2.17195473172016193047075953608, −1.45171914331201204100411681819, 1.13781267947280942873602575313, 2.20542201339905660551082965334, 3.33694906993858872358013293080, 4.23909042007518049012742607455, 4.74155860173287316585038302574, 6.22653832540350234916996244928, 6.84351603969562820459861685075, 7.77673910156982210587742926844, 8.615824850496839592892370746025, 9.115464445602011562576777351904

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