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Properties

Label 2-15e2-45.34-c3-0-37
Degree $2$
Conductor $225$
Sign $-0.641 + 0.767i$
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  + (−2.92 − 1.68i)2-s + (4.97 − 1.5i)3-s + (1.68 + 2.92i)4-s + (−17.0 − 4.00i)6-s + (1.40 + 0.813i)7-s + 15.6i·8-s + (22.5 − 14.9i)9-s + (16.4 − 28.4i)11-s + (12.7 + 12i)12-s + (−28.5 + 16.5i)13-s + (−2.74 − 4.75i)14-s + (39.8 − 68.9i)16-s − 110. i·17-s + (−90.8 + 5.64i)18-s + 54.3·19-s + ⋯
L(s)  = 1  + (−1.03 − 0.596i)2-s + (0.957 − 0.288i)3-s + (0.210 + 0.365i)4-s + (−1.16 − 0.272i)6-s + (0.0761 + 0.0439i)7-s + 0.689i·8-s + (0.833 − 0.552i)9-s + (0.450 − 0.780i)11-s + (0.307 + 0.288i)12-s + (−0.610 + 0.352i)13-s + (−0.0523 − 0.0907i)14-s + (0.621 − 1.07i)16-s − 1.57i·17-s + (−1.18 + 0.0739i)18-s + 0.655·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.641 + 0.767i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.641 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.527418 - 1.12790i\)
\(L(\frac12)\) \(\approx\) \(0.527418 - 1.12790i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.97 + 1.5i)T \)
5 \( 1 \)
good2 \( 1 + (2.92 + 1.68i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-1.40 - 0.813i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-16.4 + 28.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (28.5 - 16.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 110. iT - 4.91e3T^{2} \)
19 \( 1 - 54.3T + 6.85e3T^{2} \)
23 \( 1 + (58.4 - 33.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-137. + 237. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 347. iT - 5.06e4T^{2} \)
41 \( 1 + (145. + 252. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (174. + 100. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (417. + 241. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 175. iT - 1.48e5T^{2} \)
59 \( 1 + (91.6 + 158. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (218. - 378. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-720. + 415. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 118.T + 3.57e5T^{2} \)
73 \( 1 - 183. iT - 3.89e5T^{2} \)
79 \( 1 + (319. - 552. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-1.29e3 - 747. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.43e3T + 7.04e5T^{2} \)
97 \( 1 + (772. + 445. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.58147307630584497385788162010, −10.01360468979131133174246794373, −9.586155695934880575766459529086, −8.606651030048344934606478626700, −7.86704475689999814330526661396, −6.70918770154266296237394858480, −4.98808806074119403411220826660, −3.26105161529071445484340379525, −2.09949758285996934477747988260, −0.67918816010795960701625069500, 1.59446258866937769319337507776, 3.39629351976193073771249994748, 4.63182455422714472819024824361, 6.46508178291892730890520774225, 7.51739206292639073292288600136, 8.195111175414303547270152217383, 9.116084280081246681532902992804, 9.892762463441456502971894436461, 10.62062957514571043382719553866, 12.37621767305602412058386701067

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