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Properties

Label 2-15e2-5.3-c2-0-7
Degree $2$
Conductor $225$
Sign $0.850 - 0.525i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)2-s + 1.00i·4-s + (7.34 − 7.34i)7-s + (−6.12 − 6.12i)8-s + 18·11-s + (−7.34 − 7.34i)13-s + 18i·14-s + 10.9·16-s + (4.89 − 4.89i)17-s + 10i·19-s + (−22.0 + 22.0i)22-s + (19.5 + 19.5i)23-s + 18·26-s + (7.34 + 7.34i)28-s + 22·31-s + (11.0 − 11.0i)32-s + ⋯
L(s)  = 1  + (−0.612 + 0.612i)2-s + 0.250i·4-s + (1.04 − 1.04i)7-s + (−0.765 − 0.765i)8-s + 1.63·11-s + (−0.565 − 0.565i)13-s + 1.28i·14-s + 0.687·16-s + (0.288 − 0.288i)17-s + 0.526i·19-s + (−1.00 + 1.00i)22-s + (0.851 + 0.851i)23-s + 0.692·26-s + (0.262 + 0.262i)28-s + 0.709·31-s + (0.344 − 0.344i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.24137 + 0.352649i\)
\(L(\frac12)\) \(\approx\) \(1.24137 + 0.352649i\)
\(L(2)\) 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 \( 1 \)
good2 \( 1 + (1.22 - 1.22i)T - 4iT^{2} \)
7 \( 1 + (-7.34 + 7.34i)T - 49iT^{2} \)
11 \( 1 - 18T + 121T^{2} \)
13 \( 1 + (7.34 + 7.34i)T + 169iT^{2} \)
17 \( 1 + (-4.89 + 4.89i)T - 289iT^{2} \)
19 \( 1 - 10iT - 361T^{2} \)
23 \( 1 + (-19.5 - 19.5i)T + 529iT^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 - 22T + 961T^{2} \)
37 \( 1 + (-7.34 + 7.34i)T - 1.36e3iT^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 + (-29.3 - 29.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (44.0 - 44.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (4.89 + 4.89i)T + 2.80e3iT^{2} \)
59 \( 1 + 90iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (-44.0 + 44.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 72T + 5.04e3T^{2} \)
73 \( 1 + (44.0 + 44.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 70iT - 6.24e3T^{2} \)
83 \( 1 + (53.8 + 53.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 90iT - 7.92e3T^{2} \)
97 \( 1 + (102. - 102. i)T - 9.40e3iT^{2} \)
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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

−11.96541902600699185137935873534, −11.18773608184260338082962175803, −9.913827481606088443253229516651, −9.063006450344345750711235521767, −7.910834889814330713313011947675, −7.35933896262784733371094042540, −6.27880921522558439170328361108, −4.64495861664879445859095010401, −3.47329177377629077118638421789, −1.12239581875821664785303519547, 1.32098563901636395253608794760, 2.52225572703026884565144681282, 4.49705598293395536691563928818, 5.66984499577929574490749411901, 6.85035907410755275183033147125, 8.496339626870614057560859018177, 8.984529558874906902470964877827, 9.909974764137690431233296653533, 11.14937838924044293543513859599, 11.69993336690140683394487889008

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