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Properties

Label 2-245-7.4-c3-0-4
Degree $2$
Conductor $245$
Sign $-0.991 + 0.126i$
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  + (1.15 + 2.00i)2-s + (−2.5 + 4.33i)3-s + (1.31 − 2.28i)4-s + (−2.5 − 4.33i)5-s − 11.5·6-s + 24.6·8-s + (0.999 + 1.73i)9-s + (5.79 − 10.0i)10-s + (−23.1 + 40.0i)11-s + (6.58 + 11.4i)12-s − 61.3·13-s + 25.0·15-s + (18 + 31.1i)16-s + (−50.6 + 87.7i)17-s + (−2.31 + 4.01i)18-s + (−1.83 − 3.17i)19-s + ⋯
L(s)  = 1  + (0.409 + 0.709i)2-s + (−0.481 + 0.833i)3-s + (0.164 − 0.285i)4-s + (−0.223 − 0.387i)5-s − 0.788·6-s + 1.08·8-s + (0.0370 + 0.0641i)9-s + (0.183 − 0.317i)10-s + (−0.634 + 1.09i)11-s + (0.158 + 0.274i)12-s − 1.30·13-s + 0.430·15-s + (0.281 + 0.487i)16-s + (−0.722 + 1.25i)17-s + (−0.0303 + 0.0525i)18-s + (−0.0221 − 0.0383i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 245,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0685004 - 1.07942i\)
\(L(\frac12)\) \(\approx\) \(0.0685004 - 1.07942i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.15 - 2.00i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (23.1 - 40.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 61.3T + 2.19e3T^{2} \)
17 \( 1 + (50.6 - 87.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (1.83 + 3.17i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42.4 + 73.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 30.1T + 2.43e4T^{2} \)
31 \( 1 + (94.4 - 163. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (9.03 + 15.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 481.T + 6.89e4T^{2} \)
43 \( 1 + 97.7T + 7.95e4T^{2} \)
47 \( 1 + (58.8 + 101. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (333. - 578. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (28.6 - 49.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (369. + 639. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (276. - 478. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 740.T + 3.57e5T^{2} \)
73 \( 1 + (116. - 202. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-537. - 931. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 683.T + 5.71e5T^{2} \)
89 \( 1 + (-690. - 1.19e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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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

−12.30606631603374912053860792002, −10.87922356280042905916743603743, −10.37024256727599867117520923424, −9.465899171146371592112603973610, −7.945012703827366374742883711340, −7.08880741765671910998399862954, −5.84521017209001430072368745616, −4.79333828501458035574750507644, −4.38935713784734530194269727193, −2.04219889507036837077853262329, 0.37261880346719232517167380483, 2.18898454662478548698924394158, 3.25313386057147293321998409899, 4.70394493366525851689294643254, 6.07431957106300509556175510381, 7.30092563482984294736755038543, 7.73987482626082993943517757335, 9.381259794473713899313097365392, 10.61235056767677837919039114192, 11.53102472339208413001834271240

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