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Properties

Label 2-245-7.2-c3-0-28
Degree $2$
Conductor $245$
Sign $0.827 + 0.561i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
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  + (−0.828 + 1.43i)2-s + (0.166 + 0.288i)3-s + (2.62 + 4.55i)4-s + (2.5 − 4.33i)5-s − 0.551·6-s − 21.9·8-s + (13.4 − 23.2i)9-s + (4.14 + 7.17i)10-s + (−34.7 − 60.2i)11-s + (−0.875 + 1.51i)12-s + 68.4·13-s + 1.66·15-s + (−2.85 + 4.93i)16-s + (−52.1 − 90.3i)17-s + (22.2 + 38.5i)18-s + (−35.9 + 62.2i)19-s + ⋯
L(s)  = 1  + (−0.292 + 0.507i)2-s + (0.0320 + 0.0554i)3-s + (0.328 + 0.569i)4-s + (0.223 − 0.387i)5-s − 0.0375·6-s − 0.970·8-s + (0.497 − 0.862i)9-s + (0.130 + 0.226i)10-s + (−0.953 − 1.65i)11-s + (−0.0210 + 0.0364i)12-s + 1.45·13-s + 0.0286·15-s + (−0.0445 + 0.0771i)16-s + (−0.744 − 1.28i)17-s + (0.291 + 0.504i)18-s + (−0.434 + 0.751i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.39609 - 0.428977i\)
\(L(\frac12)\) \(\approx\) \(1.39609 - 0.428977i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 \)
good2 \( 1 + (0.828 - 1.43i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-0.166 - 0.288i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (34.7 + 60.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 68.4T + 2.19e3T^{2} \)
17 \( 1 + (52.1 + 90.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (35.9 - 62.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-50.5 + 87.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 114.T + 2.43e4T^{2} \)
31 \( 1 + (-36.8 - 63.7i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-100. + 174. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 417.T + 6.89e4T^{2} \)
43 \( 1 - 311.T + 7.95e4T^{2} \)
47 \( 1 + (74.8 - 129. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (135. + 235. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-259. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-109. + 190. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (40.3 + 69.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 91.0T + 3.57e5T^{2} \)
73 \( 1 + (441. + 764. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (299. - 519. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 70.8T + 5.71e5T^{2} \)
89 \( 1 + (401. - 695. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 145.T + 9.12e5T^{2} \)
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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

−11.43342204922436063232660861626, −10.76161861276569300064908905122, −9.187352335151299417941215517868, −8.667038898772841344872883611726, −7.70128718883956090707370152322, −6.45718042176452411340380899956, −5.74632463633341633607194806838, −3.97344469780258031119563783453, −2.82147425881325501850829444401, −0.64412351731685371685519377237, 1.59497003279133665886474973402, 2.48920818609778378440193328432, 4.31664291935128267601362511119, 5.65206899908154382370048976536, 6.72687809012683983600073792859, 7.76832659139287965215725209096, 9.072802180890661767186741900757, 10.09068286966676660992752762666, 10.74408814033037973328368769223, 11.32213965450492949227349512449

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