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Properties

Label 2-245-7.4-c3-0-18
Degree $2$
Conductor $245$
Sign $-0.605 - 0.795i$
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  + (2.20 + 3.82i)2-s + (1.62 − 2.80i)3-s + (−5.74 + 9.94i)4-s + (2.5 + 4.33i)5-s + 14.3·6-s − 15.3·8-s + (8.24 + 14.2i)9-s + (−11.0 + 19.1i)10-s + (0.0710 − 0.123i)11-s + (18.6 + 32.2i)12-s + 32.1·13-s + 16.2·15-s + (11.9 + 20.7i)16-s + (−57.1 + 99.0i)17-s + (−36.3 + 63.0i)18-s + (21.6 + 37.4i)19-s + ⋯
L(s)  = 1  + (0.780 + 1.35i)2-s + (0.312 − 0.540i)3-s + (−0.717 + 1.24i)4-s + (0.223 + 0.387i)5-s + 0.973·6-s − 0.679·8-s + (0.305 + 0.528i)9-s + (−0.348 + 0.604i)10-s + (0.00194 − 0.00337i)11-s + (0.447 + 0.775i)12-s + 0.685·13-s + 0.279·15-s + (0.187 + 0.324i)16-s + (−0.815 + 1.41i)17-s + (−0.476 + 0.825i)18-s + (0.260 + 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 245,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38106 + 2.78593i\)
\(L(\frac12)\) \(\approx\) \(1.38106 + 2.78593i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.20 - 3.82i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.62 + 2.80i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-0.0710 + 0.123i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 32.1T + 2.19e3T^{2} \)
17 \( 1 + (57.1 - 99.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-21.6 - 37.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (77.2 + 133. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 40.1T + 2.43e4T^{2} \)
31 \( 1 + (-37.7 + 65.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-200. - 346. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 95.4T + 6.89e4T^{2} \)
43 \( 1 + 340.T + 7.95e4T^{2} \)
47 \( 1 + (3.74 + 6.48i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-338. + 586. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-398. + 689. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (378. + 655. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-370. + 641. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 37.0T + 3.57e5T^{2} \)
73 \( 1 + (-40.4 + 70.0i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-158. - 274. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 945.T + 5.71e5T^{2} \)
89 \( 1 + (-391. - 678. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 393.T + 9.12e5T^{2} \)
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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

−12.53972724038406608633053267456, −11.04969690517699536475888900812, −10.06882349500950981293192716182, −8.348653906385180095266010858707, −8.007598646553075683953180385851, −6.65117884263038200576685468155, −6.26247150067001374426035479480, −4.89369324706849392765637473879, −3.74448330777780962927526984765, −1.93037138407507191390513118077, 1.01865867376247264992252096162, 2.54767875780295180256588556270, 3.72671461965347401381460901201, 4.56201110594098337752965509275, 5.69748808874565611836903659252, 7.28332542740173970923112638295, 8.966128187071077809391244786368, 9.545948062889240562333789248288, 10.49617108060004549691954252104, 11.48151012973498408543724822191

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