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Properties

Label 2-15e2-15.2-c3-0-14
Degree $2$
Conductor $225$
Sign $0.391 + 0.920i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.62 − 2.62i)2-s − 5.74i·4-s + (10.8 + 10.8i)7-s + (5.91 + 5.91i)8-s − 37.8i·11-s + (48.1 − 48.1i)13-s + 56.9·14-s + 76.9·16-s + (60.3 − 60.3i)17-s + 109. i·19-s + (−99.3 − 99.3i)22-s + (−39.7 − 39.7i)23-s − 252. i·26-s + (62.4 − 62.4i)28-s + 90.3·29-s + ⋯
L(s)  = 1  + (0.926 − 0.926i)2-s − 0.717i·4-s + (0.586 + 0.586i)7-s + (0.261 + 0.261i)8-s − 1.03i·11-s + (1.02 − 1.02i)13-s + 1.08·14-s + 1.20·16-s + (0.860 − 0.860i)17-s + 1.32i·19-s + (−0.962 − 0.962i)22-s + (−0.360 − 0.360i)23-s − 1.90i·26-s + (0.421 − 0.421i)28-s + 0.578·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.74527 - 1.81643i\)
\(L(\frac12)\) \(\approx\) \(2.74527 - 1.81643i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.62 + 2.62i)T - 8iT^{2} \)
7 \( 1 + (-10.8 - 10.8i)T + 343iT^{2} \)
11 \( 1 + 37.8iT - 1.33e3T^{2} \)
13 \( 1 + (-48.1 + 48.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-60.3 + 60.3i)T - 4.91e3iT^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 + (39.7 + 39.7i)T + 1.21e4iT^{2} \)
29 \( 1 - 90.3T + 2.43e4T^{2} \)
31 \( 1 + 233.T + 2.97e4T^{2} \)
37 \( 1 + (-19.0 - 19.0i)T + 5.06e4iT^{2} \)
41 \( 1 - 260. iT - 6.89e4T^{2} \)
43 \( 1 + (176. - 176. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-145. + 145. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-183. - 183. i)T + 1.48e5iT^{2} \)
59 \( 1 - 279.T + 2.05e5T^{2} \)
61 \( 1 + 390.T + 2.26e5T^{2} \)
67 \( 1 + (150. + 150. i)T + 3.00e5iT^{2} \)
71 \( 1 + 470. iT - 3.57e5T^{2} \)
73 \( 1 + (480. - 480. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.32e3iT - 4.93e5T^{2} \)
83 \( 1 + (456. + 456. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (-785. - 785. i)T + 9.12e5iT^{2} \)
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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

−11.70590102939686350540432575795, −10.96976876622391924353888372490, −10.08610435316090518829572533231, −8.533493000592066893122451307525, −7.86454903655888390680623538138, −5.91058942008442019936408549243, −5.26552677814666436265864695409, −3.78309046010550123818256756626, −2.87651144190281946613445622995, −1.30226608592513718068294178292, 1.54523838352609036660662393498, 3.81790333195009509382033913405, 4.60974956537953200016972744137, 5.74669515497610561814147047439, 6.88114317760709615597554703575, 7.54374750262491625403293872097, 8.826618439942168706728126849306, 10.12176667202303660385697845189, 11.08446044712466221679104675532, 12.24276156082658951948412199834

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