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Properties

Label 2-245-35.12-c1-0-11
Degree $2$
Conductor $245$
Sign $-0.470 + 0.882i$
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  + (−1.86 + 0.5i)2-s + (0.5 − 1.86i)3-s + (1.5 − 0.866i)4-s + (0.133 − 2.23i)5-s + 3.73i·6-s + (0.366 − 0.366i)8-s + (−0.633 − 0.366i)9-s + (0.866 + 4.23i)10-s + (−0.366 − 0.633i)11-s + (−0.866 − 3.23i)12-s + (−2 − 2i)13-s + (−4.09 − 1.36i)15-s + (−2.23 + 3.86i)16-s + (−1 − 0.267i)17-s + (1.36 + 0.366i)18-s + (−1.36 + 2.36i)19-s + ⋯
L(s)  = 1  + (−1.31 + 0.353i)2-s + (0.288 − 1.07i)3-s + (0.750 − 0.433i)4-s + (0.0599 − 0.998i)5-s + 1.52i·6-s + (0.129 − 0.129i)8-s + (−0.211 − 0.122i)9-s + (0.273 + 1.33i)10-s + (−0.110 − 0.191i)11-s + (−0.249 − 0.933i)12-s + (−0.554 − 0.554i)13-s + (−1.05 − 0.352i)15-s + (−0.558 + 0.966i)16-s + (−0.242 − 0.0649i)17-s + (0.321 + 0.0862i)18-s + (−0.313 + 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307188 - 0.512121i\)
\(L(\frac12)\) \(\approx\) \(0.307188 - 0.512121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (1.86 - 0.5i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.5 + 1.86i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (1 + 0.267i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.36 - 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 + 6.96i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (0.464 - 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 - 5.83i)T - 43iT^{2} \)
47 \( 1 + (0.169 + 0.633i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.83 - 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.09 - 1.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.33 - 4.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.303 - 1.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (0.928 - 3.46i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.83 - 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 - 3.09i)T + 83iT^{2} \)
89 \( 1 + (-8.33 + 14.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.92 + 7.92i)T - 97iT^{2} \)
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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

−11.89647484328690796525331316578, −10.45968658718075839087730792481, −9.644721566420751406071345485423, −8.475168000469860911523449851948, −8.120476242988960430697599433712, −7.17122927069378958133034539429, −6.08106135955873524882917946456, −4.46865246690969969199442214683, −2.13547553519354470592275621791, −0.71389620344308966359474613873, 2.19683404618992609901026736825, 3.62737802420282452233745761757, 5.00609288249513974887296413336, 6.77622377285187602716424416509, 7.70027515690924379740748435776, 8.938598668038996038116916410478, 9.645345979668662386716252448909, 10.24477203432512269242006105113, 11.02593097172270720657280324016, 11.82495470101023746995367036469

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