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Properties

Label 2-525-15.2-c1-0-20
Degree $2$
Conductor $525$
Sign $0.533 - 0.845i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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  + (−1.90 + 1.90i)2-s + (1.68 + 0.394i)3-s − 5.22i·4-s + (−3.95 + 2.45i)6-s + (0.707 + 0.707i)7-s + (6.13 + 6.13i)8-s + (2.68 + 1.33i)9-s − 3.76i·11-s + (2.06 − 8.81i)12-s + (3.48 − 3.48i)13-s − 2.68·14-s − 12.8·16-s + (0.131 − 0.131i)17-s + (−7.64 + 2.57i)18-s − 3.89i·19-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (0.973 + 0.227i)3-s − 2.61i·4-s + (−1.61 + 1.00i)6-s + (0.267 + 0.267i)7-s + (2.16 + 2.16i)8-s + (0.896 + 0.443i)9-s − 1.13i·11-s + (0.595 − 2.54i)12-s + (0.965 − 0.965i)13-s − 0.718·14-s − 3.21·16-s + (0.0319 − 0.0319i)17-s + (−1.80 + 0.608i)18-s − 0.893i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961252 + 0.530382i\)
\(L(\frac12)\) \(\approx\) \(0.961252 + 0.530382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.394i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.90 - 1.90i)T - 2iT^{2} \)
11 \( 1 + 3.76iT - 11T^{2} \)
13 \( 1 + (-3.48 + 3.48i)T - 13iT^{2} \)
17 \( 1 + (-0.131 + 0.131i)T - 17iT^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + (3.35 + 3.35i)T + 23iT^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 + (-4.98 - 4.98i)T + 37iT^{2} \)
41 \( 1 - 1.16iT - 41T^{2} \)
43 \( 1 + (2.05 - 2.05i)T - 43iT^{2} \)
47 \( 1 + (7.97 - 7.97i)T - 47iT^{2} \)
53 \( 1 + (-3.80 - 3.80i)T + 53iT^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 + 4.11T + 61T^{2} \)
67 \( 1 + (0.153 + 0.153i)T + 67iT^{2} \)
71 \( 1 + 2.12iT - 71T^{2} \)
73 \( 1 + (9.79 - 9.79i)T - 73iT^{2} \)
79 \( 1 + 0.147iT - 79T^{2} \)
83 \( 1 + (2.58 + 2.58i)T + 83iT^{2} \)
89 \( 1 + 1.17T + 89T^{2} \)
97 \( 1 + (-1.52 - 1.52i)T + 97iT^{2} \)
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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

−10.53185753857798349612677949112, −9.822667386125027675479813004214, −8.811571513771546904694267321570, −8.357896102336907890739317972269, −7.83751396025231475532607137814, −6.61465216832149459873437592154, −5.83279124309004345596915198682, −4.63325268686812337958234332183, −2.86323322536695654591179669285, −1.08485744853113726926145062942, 1.43684710656509639337575451789, 2.19274389226162268560081421056, 3.57446047791374903999003253795, 4.28666701039952396777015653078, 6.70684705521112354753204223256, 7.62171434685086213421783024742, 8.298474445665929702655251366454, 9.066625064839098201489518411430, 9.877051739090819010723149328849, 10.37276775956330758511738054940

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