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Properties

Label 2-15e2-15.2-c3-0-17
Degree $2$
Conductor $225$
Sign $-0.998 + 0.0618i$
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  + (3.74 − 3.74i)2-s − 20.0i·4-s + (−1.80 − 1.80i)7-s + (−45.3 − 45.3i)8-s − 46.0i·11-s + (−18.6 + 18.6i)13-s − 13.5·14-s − 178.·16-s + (−14.5 + 14.5i)17-s − 74.4i·19-s + (−172. − 172. i)22-s + (59.6 + 59.6i)23-s + 139. i·26-s + (−36.3 + 36.3i)28-s + 202.·29-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s − 2.51i·4-s + (−0.0977 − 0.0977i)7-s + (−2.00 − 2.00i)8-s − 1.26i·11-s + (−0.397 + 0.397i)13-s − 0.258·14-s − 2.79·16-s + (−0.208 + 0.208i)17-s − 0.899i·19-s + (−1.67 − 1.67i)22-s + (0.540 + 0.540i)23-s + 1.05i·26-s + (−0.245 + 0.245i)28-s + 1.29·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
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: \((2,\ 225,\ (\ :3/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0962923 - 3.10996i\)
\(L(\frac12)\) \(\approx\) \(0.0962923 - 3.10996i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.74 + 3.74i)T - 8iT^{2} \)
7 \( 1 + (1.80 + 1.80i)T + 343iT^{2} \)
11 \( 1 + 46.0iT - 1.33e3T^{2} \)
13 \( 1 + (18.6 - 18.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (14.5 - 14.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 74.4iT - 6.85e3T^{2} \)
23 \( 1 + (-59.6 - 59.6i)T + 1.21e4iT^{2} \)
29 \( 1 - 202.T + 2.43e4T^{2} \)
31 \( 1 + 49.5T + 2.97e4T^{2} \)
37 \( 1 + (-45.0 - 45.0i)T + 5.06e4iT^{2} \)
41 \( 1 - 306. iT - 6.89e4T^{2} \)
43 \( 1 + (-230. + 230. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-176. + 176. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-85.1 - 85.1i)T + 1.48e5iT^{2} \)
59 \( 1 - 330.T + 2.05e5T^{2} \)
61 \( 1 - 678.T + 2.26e5T^{2} \)
67 \( 1 + (756. + 756. i)T + 3.00e5iT^{2} \)
71 \( 1 - 100. iT - 3.57e5T^{2} \)
73 \( 1 + (586. - 586. i)T - 3.89e5iT^{2} \)
79 \( 1 - 286. iT - 4.93e5T^{2} \)
83 \( 1 + (-947. - 947. i)T + 5.71e5iT^{2} \)
89 \( 1 - 688.T + 7.04e5T^{2} \)
97 \( 1 + (920. + 920. i)T + 9.12e5iT^{2} \)
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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

−11.42580120790952588182627574520, −10.78707091365051295929412323313, −9.776552956449800098361139107977, −8.687719699888572171143851848699, −6.81531419982414648806161505109, −5.71426789104380861367450085462, −4.69584095232736864395392268517, −3.53840430114179321270488573225, −2.48963329746032895675964270330, −0.855123952330764660102799670898, 2.67227920963814486856359467335, 4.12904658527716359467371159150, 4.99358141958485260258351442908, 6.06973480499400735093525933959, 7.08484869311437010274597198754, 7.80639942030658494603589618942, 8.991306523191797285780427236766, 10.35620743800867504114469407524, 11.93965300129028314111124312926, 12.52120272023255630301232710219

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