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Properties

Label 2-245-35.4-c1-0-10
Degree $2$
Conductor $245$
Sign $0.962 - 0.271i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (2.44 + 1.41i)3-s + (−0.500 + 0.866i)4-s + (−0.448 − 2.19i)5-s + 2.82·6-s + 3i·8-s + (2.49 + 4.33i)9-s + (−1.48 − 1.67i)10-s + (−2.44 + 1.41i)12-s − 4.24i·13-s + (2 − 5.99i)15-s + (0.500 + 0.866i)16-s + (−3.67 − 2.12i)17-s + (4.33 + 2.5i)18-s + (1.41 + 2.44i)19-s + (2.12 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.41 + 0.816i)3-s + (−0.250 + 0.433i)4-s + (−0.200 − 0.979i)5-s + 1.15·6-s + 1.06i·8-s + (0.833 + 1.44i)9-s + (−0.469 − 0.529i)10-s + (−0.707 + 0.408i)12-s − 1.17i·13-s + (0.516 − 1.54i)15-s + (0.125 + 0.216i)16-s + (−0.891 − 0.514i)17-s + (1.02 + 0.589i)18-s + (0.324 + 0.561i)19-s + (0.474 + 0.158i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20239 + 0.304163i\)
\(L(\frac12)\) \(\approx\) \(2.20239 + 0.304163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.448 + 2.19i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-2.44 - 1.41i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + (3.67 + 2.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.19 + 3i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.92 + 4i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (11.0 + 6.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.48iT - 83T^{2} \)
89 \( 1 + (-2.12 - 3.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.24iT - 97T^{2} \)
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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

−12.37977353988219624470065449624, −11.33265278059940207005371497732, −9.969479765436604935495522424797, −9.169864961539175430110744365320, −8.259211887974300560475467237654, −7.79854165026498520128286149769, −5.44412810668871832723278910550, −4.40802441655419873847173313642, −3.63532866849199310705853611054, −2.46762104220079143664097961321, 1.99296762216752513172282824167, 3.35755073767409733152042992505, 4.48810055562412327469329330369, 6.43064286418123012046063774365, 6.82698723276605308888962268806, 7.994189159490589946696554177023, 9.042500187070389005217891172872, 9.922935020890104483901921803551, 11.17101314588107632546454601131, 12.41648181675204245904693330597

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