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Properties

Label 2-525-15.8-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.317 - 0.948i$
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  + (−1.78 − 1.78i)2-s + (0.912 + 1.47i)3-s + 4.37i·4-s + (1 − 4.25i)6-s + (−0.707 + 0.707i)7-s + (4.23 − 4.23i)8-s + (−1.33 + 2.68i)9-s + 5.84i·11-s + (−6.43 + 3.98i)12-s + (−2.38 − 2.38i)13-s + 2.52·14-s − 6.37·16-s + (−3.00 − 3.00i)17-s + (7.17 − 2.41i)18-s − 4i·19-s + ⋯
L(s)  = 1  + (−1.26 − 1.26i)2-s + (0.526 + 0.850i)3-s + 2.18i·4-s + (0.408 − 1.73i)6-s + (−0.267 + 0.267i)7-s + (1.49 − 1.49i)8-s + (−0.445 + 0.895i)9-s + 1.76i·11-s + (−1.85 + 1.15i)12-s + (−0.661 − 0.661i)13-s + 0.674·14-s − 1.59·16-s + (−0.729 − 0.729i)17-s + (1.69 − 0.568i)18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226299 + 0.314327i\)
\(L(\frac12)\) \(\approx\) \(0.226299 + 0.314327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.912 - 1.47i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.78 + 1.78i)T + 2iT^{2} \)
11 \( 1 - 5.84iT - 11T^{2} \)
13 \( 1 + (2.38 + 2.38i)T + 13iT^{2} \)
17 \( 1 + (3.00 + 3.00i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.44 - 2.44i)T - 23iT^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (4.76 - 4.76i)T - 37iT^{2} \)
41 \( 1 + 5.34iT - 41T^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 + (-5.45 - 5.45i)T + 47iT^{2} \)
53 \( 1 + (3.77 - 3.77i)T - 53iT^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + (-0.887 + 0.887i)T - 67iT^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (-0.887 - 0.887i)T + 73iT^{2} \)
79 \( 1 + 2.11iT - 79T^{2} \)
83 \( 1 + (6.93 - 6.93i)T - 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + (-2.38 + 2.38i)T - 97iT^{2} \)
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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

−10.80959801997913363917981411430, −10.12268008781056990445841977671, −9.344194988708356681605474325817, −9.109284708518591676955350407367, −7.81273351194445973358943167941, −7.19235678015918371809543218535, −5.12517390755023237500989324910, −4.07770134243584368684424930494, −2.82267979740617575652465765451, −2.08169999500637735908878520054, 0.30361197527115177067627664733, 1.88589070510322241184628102746, 3.69406089735706782308364066231, 5.67486790669410231827592046179, 6.29261998700749914166389385657, 7.13943428068634869464173397504, 7.921042450391813318855434420399, 8.706499849649198489078583710797, 9.172445007610367736679704638503, 10.31230559421691094909623835622

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