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Properties

Label 2-245-7.4-c3-0-37
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  + (0.184 + 0.319i)2-s + (4.87 − 8.43i)3-s + (3.93 − 6.81i)4-s + (−2.5 − 4.33i)5-s + 3.59·6-s + 5.85·8-s + (−33.9 − 58.7i)9-s + (0.923 − 1.59i)10-s + (−15.3 + 26.5i)11-s + (−38.2 − 66.3i)12-s + 36.4·13-s − 48.7·15-s + (−30.3 − 52.6i)16-s + (39.8 − 69.0i)17-s + (12.5 − 21.7i)18-s + (76.2 + 131. i)19-s + ⋯
L(s)  = 1  + (0.0652 + 0.113i)2-s + (0.937 − 1.62i)3-s + (0.491 − 0.851i)4-s + (−0.223 − 0.387i)5-s + 0.244·6-s + 0.258·8-s + (−1.25 − 2.17i)9-s + (0.0291 − 0.0505i)10-s + (−0.419 + 0.727i)11-s + (−0.921 − 1.59i)12-s + 0.777·13-s − 0.838·15-s + (−0.474 − 0.821i)16-s + (0.568 − 0.985i)17-s + (0.164 − 0.284i)18-s + (0.920 + 1.59i)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

\(L(2)\) \(\approx\) \(0.775002 - 2.52222i\)
\(L(\frac12)\) \(\approx\) \(0.775002 - 2.52222i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.184 - 0.319i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.87 + 8.43i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (15.3 - 26.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 + (-39.8 + 69.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-76.2 - 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (11.1 + 19.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 101.T + 2.43e4T^{2} \)
31 \( 1 + (124. - 216. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (3.77 + 6.54i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 + 237.T + 7.95e4T^{2} \)
47 \( 1 + (-165. - 286. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-243. + 422. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-358. + 621. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (177. + 307. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (28.7 - 49.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 696.T + 3.57e5T^{2} \)
73 \( 1 + (-130. + 226. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (135. + 234. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 681.T + 5.71e5T^{2} \)
89 \( 1 + (80.3 + 139. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 167.T + 9.12e5T^{2} \)
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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

−11.63836172048306306518228581853, −10.18092581095857053008642093368, −9.149977939699818037328511389998, −8.012361921828232943060530833798, −7.35181378999832911883002805180, −6.41092709776254697227068573327, −5.31045402592960656617381807310, −3.27428013739451764448903820790, −1.89765610518473799055258843024, −0.966305826235695927128100965117, 2.63752334519992308198870162755, 3.41594052557746480207952033706, 4.24928675154637587004009437753, 5.71111763303398198833108049858, 7.41075175958995660972364757338, 8.343792557292965132541204053394, 9.011501773725429642450815811042, 10.25327219218239873841244301103, 10.96298143687414338329955100596, 11.65214939772520451633599032925

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