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Properties

Label 2-245-35.4-c3-0-40
Degree $2$
Conductor $245$
Sign $0.667 + 0.744i$
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  + (1.45 − 0.837i)2-s + (2.15 + 1.24i)3-s + (−2.59 + 4.49i)4-s + (4.78 − 10.1i)5-s + 4.17·6-s + 22.1i·8-s + (−10.3 − 17.9i)9-s + (−1.51 − 18.6i)10-s + (28.7 − 49.8i)11-s + (−11.2 + 6.47i)12-s − 45.5i·13-s + (22.9 − 15.8i)15-s + (−2.26 − 3.92i)16-s + (79.6 + 46.0i)17-s + (−30.1 − 17.4i)18-s + (62.5 + 108. i)19-s + ⋯
L(s)  = 1  + (0.512 − 0.296i)2-s + (0.415 + 0.239i)3-s + (−0.324 + 0.562i)4-s + (0.428 − 0.903i)5-s + 0.284·6-s + 0.976i·8-s + (−0.384 − 0.666i)9-s + (−0.0479 − 0.590i)10-s + (0.789 − 1.36i)11-s + (−0.269 + 0.155i)12-s − 0.971i·13-s + (0.394 − 0.272i)15-s + (−0.0353 − 0.0612i)16-s + (1.13 + 0.656i)17-s + (−0.394 − 0.227i)18-s + (0.755 + 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.44720 - 1.09207i\)
\(L(\frac12)\) \(\approx\) \(2.44720 - 1.09207i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.78 + 10.1i)T \)
7 \( 1 \)
good2 \( 1 + (-1.45 + 0.837i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-2.15 - 1.24i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-28.7 + 49.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 45.5iT - 2.19e3T^{2} \)
17 \( 1 + (-79.6 - 46.0i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-62.5 - 108. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-137. + 79.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 40.1T + 2.43e4T^{2} \)
31 \( 1 + (24.7 - 42.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (200. - 115. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 169.T + 6.89e4T^{2} \)
43 \( 1 + 147. iT - 7.95e4T^{2} \)
47 \( 1 + (-58.0 + 33.5i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (232. + 134. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (120. - 208. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (45.2 + 78.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (352. + 203. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 330.T + 3.57e5T^{2} \)
73 \( 1 + (473. + 273. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (12.6 + 21.9i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 376. iT - 5.71e5T^{2} \)
89 \( 1 + (-513. - 888. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 942. iT - 9.12e5T^{2} \)
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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

−11.96344431339331523815046517407, −10.60145672215465598262804560839, −9.377376374471017246610397772804, −8.590772902364836395411477046733, −8.009131886107840065061701142645, −6.04760326659453365637006189977, −5.24317069745064905644209821283, −3.75647082007332444859834077646, −3.12031977833334641411310014969, −0.997774399470299670451578348507, 1.59606316268852866908923215998, 3.04556537787732797407212134110, 4.58594460230606103138947632546, 5.56188124252649781475712308589, 6.93685537192182731156482727226, 7.30345354681160537480336113149, 9.243990223883306917782056835210, 9.594456010435997093893190263379, 10.82180316573036571675443211191, 11.78668142555297375959010503112

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