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Properties

Label 2-245-5.3-c2-0-16
Degree $2$
Conductor $245$
Sign $0.0913 - 0.995i$
Analytic cond. $6.67576$
Root an. cond. $2.58375$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.16 + 2.16i)2-s + (1.73 + 1.73i)3-s − 5.38i·4-s + (4.47 − 2.22i)5-s − 7.49·6-s + (2.99 + 2.99i)8-s − 3.00i·9-s + (−4.86 + 14.5i)10-s + 6.24·11-s + (9.31 − 9.31i)12-s + (11.7 + 11.7i)13-s + (11.6 + 3.88i)15-s + 8.56·16-s + (3.34 − 3.34i)17-s + (6.51 + 6.51i)18-s − 13.1i·19-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (0.576 + 0.576i)3-s − 1.34i·4-s + (0.895 − 0.445i)5-s − 1.24·6-s + (0.373 + 0.373i)8-s − 0.334i·9-s + (−0.486 + 1.45i)10-s + 0.567·11-s + (0.776 − 0.776i)12-s + (0.902 + 0.902i)13-s + (0.773 + 0.259i)15-s + 0.535·16-s + (0.196 − 0.196i)17-s + (0.361 + 0.361i)18-s − 0.693i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0913 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0913 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.0913 - 0.995i$
Analytic conductor: \(6.67576\)
Root analytic conductor: \(2.58375\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1),\ 0.0913 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.986093 + 0.899765i\)
\(L(\frac12)\) \(\approx\) \(0.986093 + 0.899765i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.47 + 2.22i)T \)
7 \( 1 \)
good2 \( 1 + (2.16 - 2.16i)T - 4iT^{2} \)
3 \( 1 + (-1.73 - 1.73i)T + 9iT^{2} \)
11 \( 1 - 6.24T + 121T^{2} \)
13 \( 1 + (-11.7 - 11.7i)T + 169iT^{2} \)
17 \( 1 + (-3.34 + 3.34i)T - 289iT^{2} \)
19 \( 1 + 13.1iT - 361T^{2} \)
23 \( 1 + (-25.5 - 25.5i)T + 529iT^{2} \)
29 \( 1 - 28.5iT - 841T^{2} \)
31 \( 1 + 20.3T + 961T^{2} \)
37 \( 1 + (16.5 - 16.5i)T - 1.36e3iT^{2} \)
41 \( 1 + 45.1T + 1.68e3T^{2} \)
43 \( 1 + (21.9 + 21.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-7.64 + 7.64i)T - 2.20e3iT^{2} \)
53 \( 1 + (-52.4 - 52.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 16.8iT - 3.48e3T^{2} \)
61 \( 1 + 33.5T + 3.72e3T^{2} \)
67 \( 1 + (-70.1 + 70.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (73.1 + 73.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 41.7iT - 6.24e3T^{2} \)
83 \( 1 + (-7.39 - 7.39i)T + 6.88e3iT^{2} \)
89 \( 1 + 22.1iT - 7.92e3T^{2} \)
97 \( 1 + (73.6 - 73.6i)T - 9.40e3iT^{2} \)
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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

−12.01242677028174061318389651340, −10.65477027391124976934350353889, −9.575682719232165555538893102772, −9.046036615619130271888678009180, −8.639780961535542673496214815563, −7.09665678440623744989664027532, −6.34106213555157989858088140757, −5.12577424748800312590047278951, −3.49958072784773461558753763689, −1.31184133354240846062010217699, 1.25294175339474279569853876236, 2.32469327704969950374875956705, 3.39203352303289667431324479193, 5.60450116460896625510242189084, 6.89845129349889310211236324986, 8.169724119933184398400558812379, 8.752361684194188491568230527564, 9.859688561508882106183255095329, 10.56451713042662030301564905661, 11.30638012783881043338424086633

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