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Properties

Label 2-3528-7.2-c1-0-31
Degree $2$
Conductor $3528$
Sign $0.827 + 0.561i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
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  + (0.292 − 0.507i)5-s + (−2.41 − 4.18i)11-s + 4.24·13-s + (2.29 + 3.97i)17-s + (−0.585 + 1.01i)19-s + (0.414 − 0.717i)23-s + (2.32 + 4.03i)25-s + 2.82·29-s + (−1.41 − 2.44i)31-s + (−4.82 + 8.36i)37-s − 1.75·41-s + 11.3·43-s + (6.24 − 10.8i)47-s + (−1 − 1.73i)53-s − 2.82·55-s + ⋯
L(s)  = 1  + (0.130 − 0.226i)5-s + (−0.727 − 1.26i)11-s + 1.17·13-s + (0.556 + 0.963i)17-s + (−0.134 + 0.232i)19-s + (0.0863 − 0.149i)23-s + (0.465 + 0.806i)25-s + 0.525·29-s + (−0.254 − 0.439i)31-s + (−0.793 + 1.37i)37-s − 0.274·41-s + 1.72·43-s + (0.910 − 1.57i)47-s + (−0.137 − 0.237i)53-s − 0.381·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.941939831\)
\(L(\frac12)\) \(\approx\) \(1.941939831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.41 + 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + (-2.29 - 3.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.585 - 1.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.414 + 0.717i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (1.41 + 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.82 - 8.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 + (-6.24 + 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.24 + 7.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.53 + 2.65i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.65 - 9.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.48T + 71T^{2} \)
73 \( 1 + (8.12 + 14.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-7.12 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.24T + 97T^{2} \)
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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

−8.527755922879205618358185781909, −7.966119339234392823682587344687, −7.00869708701247482390260549092, −5.98522793380414316541365399664, −5.73868039920196990961582729275, −4.73048444403179467226493724720, −3.64197867197815984920677881978, −3.12980963696560682555361046007, −1.80133418288587554870156136126, −0.74587973688497708415300487148, 0.957575385641607357981440954760, 2.22300859847493642866264710867, 2.97875517634332039104296996868, 4.06686320229573063532001233227, 4.82498845050521312602144438180, 5.64088889694106201505001935003, 6.41542599728200339349789375196, 7.32365344604382993978726729238, 7.67719265416995278031806269679, 8.783876312732036683633889088711

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