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Properties

Label 2-245-7.4-c3-0-10
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  + (−2.23 − 3.86i)2-s + (−4.90 + 8.49i)3-s + (−5.97 + 10.3i)4-s + (2.5 + 4.33i)5-s + 43.8·6-s + 17.6·8-s + (−34.6 − 59.9i)9-s + (11.1 − 19.3i)10-s + (28.2 − 48.9i)11-s + (−58.6 − 101. i)12-s + 40.9·13-s − 49.0·15-s + (8.33 + 14.4i)16-s + (−1.09 + 1.89i)17-s + (−154. + 267. i)18-s + (−8.23 − 14.2i)19-s + ⋯
L(s)  = 1  + (−0.789 − 1.36i)2-s + (−0.943 + 1.63i)3-s + (−0.747 + 1.29i)4-s + (0.223 + 0.387i)5-s + 2.98·6-s + 0.781·8-s + (−1.28 − 2.22i)9-s + (0.353 − 0.611i)10-s + (0.774 − 1.34i)11-s + (−1.41 − 2.44i)12-s + 0.873·13-s − 0.844·15-s + (0.130 + 0.225i)16-s + (−0.0156 + 0.0270i)17-s + (−2.02 + 3.50i)18-s + (−0.0994 − 0.172i)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

\(L(2)\) \(\approx\) \(0.680962 + 0.209239i\)
\(L(\frac12)\) \(\approx\) \(0.680962 + 0.209239i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.23 + 3.86i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (4.90 - 8.49i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-28.2 + 48.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 40.9T + 2.19e3T^{2} \)
17 \( 1 + (1.09 - 1.89i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8.23 + 14.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-77.6 - 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 6.26T + 2.43e4T^{2} \)
31 \( 1 + (84.3 - 146. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-18.5 - 32.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 266.T + 6.89e4T^{2} \)
43 \( 1 + 14.6T + 7.95e4T^{2} \)
47 \( 1 + (-84.9 - 147. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-75.6 + 131. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-117. + 202. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-121. - 209. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (410. - 710. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 961.T + 3.57e5T^{2} \)
73 \( 1 + (467. - 808. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (150. + 260. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + (-562. - 973. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 752.T + 9.12e5T^{2} \)
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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

−11.28694825953481109950036500648, −10.87974659927101314764251162824, −10.07395897398537514673037801945, −9.187342324274121920983865282254, −8.637191089421316055847808344727, −6.39935090074827232289309097866, −5.41143809827071394550940644437, −3.80822672294826981637972177323, −3.25400591087718546514106331528, −0.986632541498222912700778136476, 0.57058003681543517947771838396, 1.80670967955528825986142911373, 4.88745225898443405681965246276, 6.00354068497805149144697549057, 6.64594926305718322801428913751, 7.35714247013534650899701708944, 8.278483040869171906204601242104, 9.193608438524231561183434840546, 10.53032494817948864713277754726, 11.77793316128099994360034502554

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