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Properties

Label 2-245-7.2-c3-0-27
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  + (−2.15 + 3.73i)2-s + (2.5 + 4.33i)3-s + (−5.31 − 9.20i)4-s + (2.5 − 4.33i)5-s − 21.5·6-s + 11.3·8-s + (0.999 − 1.73i)9-s + (10.7 + 18.6i)10-s + (−9.86 − 17.0i)11-s + (26.5 − 46.0i)12-s − 71.3·13-s + 25.0·15-s + (17.9 − 31.1i)16-s + (−15.6 − 27.1i)17-s + (4.31 + 7.47i)18-s + (68.1 − 118. i)19-s + ⋯
L(s)  = 1  + (−0.763 + 1.32i)2-s + (0.481 + 0.833i)3-s + (−0.664 − 1.15i)4-s + (0.223 − 0.387i)5-s − 1.46·6-s + 0.502·8-s + (0.0370 − 0.0641i)9-s + (0.341 + 0.591i)10-s + (−0.270 − 0.468i)11-s + (0.639 − 1.10i)12-s − 1.52·13-s + 0.430·15-s + (0.281 − 0.487i)16-s + (−0.223 − 0.387i)17-s + (0.0565 + 0.0979i)18-s + (0.823 − 1.42i)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: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.714636 - 0.0453509i\)
\(L(\frac12)\) \(\approx\) \(0.714636 - 0.0453509i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.15 - 3.73i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (9.86 + 17.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 71.3T + 2.19e3T^{2} \)
17 \( 1 + (15.6 + 27.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-68.1 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-50.4 + 87.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 288.T + 2.43e4T^{2} \)
31 \( 1 + (104. + 180. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (154. - 268. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 181.T + 6.89e4T^{2} \)
43 \( 1 + 18.2T + 7.95e4T^{2} \)
47 \( 1 + (73.8 - 127. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-63.9 - 110. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-161. - 279. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-170. + 295. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-42.1 - 73.0i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 315.T + 3.57e5T^{2} \)
73 \( 1 + (546. + 946. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (616. - 1.06e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 643.T + 5.71e5T^{2} \)
89 \( 1 + (-570. + 987. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.41e3T + 9.12e5T^{2} \)
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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

−11.43644783679691121236575044786, −10.07064773681684615662496169505, −9.341815734282875035831362349874, −8.899230745348752695774211636221, −7.66705103341604041720063058618, −6.88389736978924424688416294339, −5.49563093099753924726975872492, −4.63193146403438664272575340892, −2.87427635825839520143647599459, −0.33844029377744374373785922247, 1.59463382226284353460502792814, 2.33229638549031083540767781806, 3.57268462936743338421946780884, 5.43998430846459827254227490325, 7.20358123533573422575507075440, 7.74625580802221601183162688047, 9.024204614898461560856985851367, 9.882420703727067488057665318087, 10.57518036608768791474306250836, 11.66828356757392019925528097138

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