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Properties

Label 2-245-7.4-c3-0-39
Degree $2$
Conductor $245$
Sign $-0.266 - 0.963i$
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  + (−1.29 − 2.23i)2-s + (3.32 − 5.76i)3-s + (0.656 − 1.13i)4-s + (−2.5 − 4.33i)5-s − 17.2·6-s − 24.0·8-s + (−8.65 − 14.9i)9-s + (−6.46 + 11.1i)10-s + (−19.1 + 33.1i)11-s + (−4.37 − 7.57i)12-s − 19.3·13-s − 33.2·15-s + (25.8 + 44.8i)16-s + (−43.6 + 75.5i)17-s + (−22.3 + 38.7i)18-s + (−22.1 − 38.3i)19-s + ⋯
L(s)  = 1  + (−0.457 − 0.791i)2-s + (0.640 − 1.10i)3-s + (0.0821 − 0.142i)4-s + (−0.223 − 0.387i)5-s − 1.17·6-s − 1.06·8-s + (−0.320 − 0.555i)9-s + (−0.204 + 0.354i)10-s + (−0.524 + 0.908i)11-s + (−0.105 − 0.182i)12-s − 0.412·13-s − 0.572·15-s + (0.404 + 0.700i)16-s + (−0.622 + 1.07i)17-s + (−0.293 + 0.507i)18-s + (−0.267 − 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 245,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.460992 + 0.605966i\)
\(L(\frac12)\) \(\approx\) \(0.460992 + 0.605966i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.29 + 2.23i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.32 + 5.76i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (19.1 - 33.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 19.3T + 2.19e3T^{2} \)
17 \( 1 + (43.6 - 75.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22.1 + 38.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (109. + 188. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 46.9T + 2.43e4T^{2} \)
31 \( 1 + (-97.2 + 168. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (183. + 317. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 339.T + 6.89e4T^{2} \)
43 \( 1 + 226.T + 7.95e4T^{2} \)
47 \( 1 + (-5.83 - 10.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-104. + 181. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (308 - 533. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-160. - 277. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (7.25 - 12.5i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + (-412. + 714. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (78.1 + 135. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + (85.1 + 147. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−10.88914056082519047653912446543, −10.10229272870236878239779795359, −8.953201790815066730016223450535, −8.171247123008942661165983142654, −7.13890115981328138917842738550, −6.05586929190493162373579043994, −4.37131770185746373263737372946, −2.50846951548437255664620881159, −1.86968840201210435871403332402, −0.30060263136164210245120520787, 2.82702218361181578424852030203, 3.67581445552852315430229999852, 5.16902821837246979806840328257, 6.47190589734275662209264548578, 7.61165272321353167302387920519, 8.404602088416747718113850526790, 9.300796866197767427322069308033, 10.09756528863387913962835844708, 11.23271891813256850529980899588, 12.11977451013579984539069099625

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