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Properties

Label 2-245-7.4-c3-0-11
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: $\chi_{245} (116, \cdot )$
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

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

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