Properties

Label 2-245-245.103-c3-0-1
Degree $2$
Conductor $245$
Sign $-0.859 + 0.511i$
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  + (−2.76 + 3.74i)2-s + (6.43 − 1.21i)3-s + (−4.01 − 13.0i)4-s + (−5.54 − 9.70i)5-s + (−13.2 + 27.4i)6-s + (−2.70 − 18.3i)7-s + (24.6 + 8.61i)8-s + (14.7 − 5.78i)9-s + (51.6 + 6.07i)10-s + (−22.8 + 58.1i)11-s + (−41.6 − 78.7i)12-s + (12.7 + 1.43i)13-s + (76.0 + 40.4i)14-s + (−47.4 − 55.7i)15-s + (−10.2 + 6.96i)16-s + (−73.9 − 2.76i)17-s + ⋯
L(s)  = 1  + (−0.975 + 1.32i)2-s + (1.23 − 0.234i)3-s + (−0.501 − 1.62i)4-s + (−0.495 − 0.868i)5-s + (−0.898 + 1.86i)6-s + (−0.146 − 0.989i)7-s + (1.08 + 0.380i)8-s + (0.546 − 0.214i)9-s + (1.63 + 0.192i)10-s + (−0.625 + 1.59i)11-s + (−1.00 − 1.89i)12-s + (0.271 + 0.0305i)13-s + (1.45 + 0.772i)14-s + (−0.816 − 0.958i)15-s + (−0.159 + 0.108i)16-s + (−1.05 − 0.0394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0456968 - 0.166275i\)
\(L(\frac12)\) \(\approx\) \(0.0456968 - 0.166275i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (5.54 + 9.70i)T \)
7 \( 1 + (2.70 + 18.3i)T \)
good2 \( 1 + (2.76 - 3.74i)T + (-2.35 - 7.64i)T^{2} \)
3 \( 1 + (-6.43 + 1.21i)T + (25.1 - 9.86i)T^{2} \)
11 \( 1 + (22.8 - 58.1i)T + (-975. - 905. i)T^{2} \)
13 \( 1 + (-12.7 - 1.43i)T + (2.14e3 + 488. i)T^{2} \)
17 \( 1 + (73.9 + 2.76i)T + (4.89e3 + 367. i)T^{2} \)
19 \( 1 + (25.7 - 44.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.39 - 90.6i)T + (-1.21e4 + 909. i)T^{2} \)
29 \( 1 + (145. - 33.2i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (99.6 - 57.5i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-140. + 74.2i)T + (2.85e4 - 4.18e4i)T^{2} \)
41 \( 1 + (75.2 + 156. i)T + (-4.29e4 + 5.38e4i)T^{2} \)
43 \( 1 + (-53.3 - 152. i)T + (-6.21e4 + 4.95e4i)T^{2} \)
47 \( 1 + (-144. - 106. i)T + (3.06e4 + 9.92e4i)T^{2} \)
53 \( 1 + (353. + 186. i)T + (8.38e4 + 1.23e5i)T^{2} \)
59 \( 1 + (2.94 + 39.2i)T + (-2.03e5 + 3.06e4i)T^{2} \)
61 \( 1 + (-277. + 900. i)T + (-1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (463. + 124. i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + (-102. + 449. i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (-501. + 370. i)T + (1.14e5 - 3.71e5i)T^{2} \)
79 \( 1 + (-359. - 207. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-131. - 1.16e3i)T + (-5.57e5 + 1.27e5i)T^{2} \)
89 \( 1 + (-5.09 - 12.9i)T + (-5.16e5 + 4.79e5i)T^{2} \)
97 \( 1 + (-181. - 181. i)T + 9.12e5iT^{2} \)
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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

−12.59828461280243977425019250275, −10.86987038358531747678827748260, −9.577833837078923224650231617820, −9.168054453443645721980485867851, −7.961597224881546618911321554881, −7.69162120846153157891894552696, −6.77154781488337161385894622908, −5.10690939103759884092667598443, −3.85109437821667916069610520306, −1.72424983667180139567658431522, 0.07724244749206817360381535661, 2.38507154408603862657403228938, 2.87827934435615108587517296402, 3.85300656878068030886498701851, 6.08628335898546304574199727177, 7.77042603802788266947309988420, 8.677699783155805487781633315637, 8.927788418547880791912922427605, 10.14886416594282797956662318750, 11.10667451719767797641767690603

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