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Properties

Label 2-245-7.3-c2-0-11
Degree $2$
Conductor $245$
Sign $0.553 - 0.832i$
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  + (0.5 + 0.866i)2-s + (3.87 + 2.23i)3-s + (1.5 − 2.59i)4-s + (−1.93 + 1.11i)5-s + 4.47i·6-s + 7·8-s + (5.5 + 9.52i)9-s + (−1.93 − 1.11i)10-s + (−1 + 1.73i)11-s + (11.6 − 6.70i)12-s + 13.4i·13-s − 10.0·15-s + (−2.5 − 4.33i)16-s + (23.2 + 13.4i)17-s + (−5.5 + 9.52i)18-s + (11.6 − 6.70i)19-s + ⋯
L(s)  = 1  + (0.250 + 0.433i)2-s + (1.29 + 0.745i)3-s + (0.375 − 0.649i)4-s + (−0.387 + 0.223i)5-s + 0.745i·6-s + 0.875·8-s + (0.611 + 1.05i)9-s + (−0.193 − 0.111i)10-s + (−0.0909 + 0.157i)11-s + (0.968 − 0.559i)12-s + 1.03i·13-s − 0.666·15-s + (−0.156 − 0.270i)16-s + (1.36 + 0.789i)17-s + (−0.305 + 0.529i)18-s + (0.611 − 0.353i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(6.67576\)
Root analytic conductor: \(2.58375\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.50865 + 1.34425i\)
\(L(\frac12)\) \(\approx\) \(2.50865 + 1.34425i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T + (-2 + 3.46i)T^{2} \)
3 \( 1 + (-3.87 - 2.23i)T + (4.5 + 7.79i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 13.4iT - 169T^{2} \)
17 \( 1 + (-23.2 - 13.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.6 + 6.70i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (13 + 22.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 22T + 841T^{2} \)
31 \( 1 + (46.4 + 26.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (7 + 12.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 26.8iT - 1.68e3T^{2} \)
43 \( 1 + 34T + 1.84e3T^{2} \)
47 \( 1 + (-23.2 + 13.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-17 + 29.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (34.8 + 20.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (81.3 - 46.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 62T + 5.04e3T^{2} \)
73 \( 1 + (46.4 + 26.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (19 + 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 40.2iT - 6.88e3T^{2} \)
89 \( 1 + (-23.2 + 13.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 26.8iT - 9.40e3T^{2} \)
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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

−11.95414607417626084186811517353, −10.83214360804393839620390521451, −9.968304557979969606118412589925, −9.195984047829484032009287717344, −8.017315210354615845561494198186, −7.19161228971948224195511734877, −5.87020734211045585980949158990, −4.52293093362063188319181736114, −3.50496026266258321000482213658, −2.00037996981255593958463649111, 1.56030048769393479063871158816, 3.04912542249195318936393808313, 3.57832550164358794703555500922, 5.43194497350807717004152102539, 7.38424908468801504469808526276, 7.61022066592897011094966974948, 8.533721753192131792451210681064, 9.681595767967163958382104116050, 10.94915266783571930576686112089, 12.10806188462632992986848167089

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