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Properties

Label 2-245-7.4-c3-0-38
Degree $2$
Conductor $245$
Sign $0.198 - 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
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: \((2,\ 245,\ (\ :3/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.885160 + 0.724192i\)
\(L(\frac12)\) \(\approx\) \(0.885160 + 0.724192i\)
\(L(\frac{5}{2})\) 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

−11.30489288078591942781061602835, −9.650877185032239113351521530769, −8.946113943840270376006813609400, −8.222560626264320296387482597876, −7.34900752801320840937869625023, −6.01546409820841443398656185084, −3.63050473997052684763499541867, −2.70653730719676634783794210357, −1.47067536513650077645961506110, −0.55414174357470051121512198478, 2.72473977623712034531686950619, 4.27974082531796482029598873144, 5.09498269250902571649449874802, 6.70216756765219716599378904399, 7.63431254407420386536463291041, 8.593901177493099175895873263716, 9.427498560375477054003865284981, 9.908999903775620491135724321718, 10.87490356735567836189543812240, 12.36558974075136686893443604021

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