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Properties

Label 2-245-7.4-c3-0-7
Degree $2$
Conductor $245$
Sign $0.991 - 0.126i$
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  + (−1.74 − 3.01i)2-s + (−0.425 + 0.736i)3-s + (−2.07 + 3.59i)4-s + (−2.5 − 4.33i)5-s + 2.96·6-s − 13.4·8-s + (13.1 + 22.7i)9-s + (−8.71 + 15.0i)10-s + (3.45 − 5.98i)11-s + (−1.76 − 3.05i)12-s − 22.1·13-s + 4.25·15-s + (39.9 + 69.2i)16-s + (−44.1 + 76.4i)17-s + (45.7 − 79.3i)18-s + (−18.4 − 32.0i)19-s + ⋯
L(s)  = 1  + (−0.616 − 1.06i)2-s + (−0.0818 + 0.141i)3-s + (−0.259 + 0.449i)4-s + (−0.223 − 0.387i)5-s + 0.201·6-s − 0.593·8-s + (0.486 + 0.842i)9-s + (−0.275 + 0.477i)10-s + (0.0946 − 0.163i)11-s + (−0.0424 − 0.0735i)12-s − 0.472·13-s + 0.0731·15-s + (0.624 + 1.08i)16-s + (−0.629 + 1.09i)17-s + (0.599 − 1.03i)18-s + (−0.223 − 0.386i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
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: \((2,\ 245,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.832724 + 0.0528448i\)
\(L(\frac12)\) \(\approx\) \(0.832724 + 0.0528448i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.74 + 3.01i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (0.425 - 0.736i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-3.45 + 5.98i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 22.1T + 2.19e3T^{2} \)
17 \( 1 + (44.1 - 76.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (18.4 + 32.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-47.7 - 82.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 269.T + 2.43e4T^{2} \)
31 \( 1 + (98.5 - 170. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (1.07 + 1.85i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 174.T + 6.89e4T^{2} \)
43 \( 1 + 17.0T + 7.95e4T^{2} \)
47 \( 1 + (-264. - 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-320. + 555. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-321. + 556. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (71.4 + 123. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (239. - 414. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 105.T + 3.57e5T^{2} \)
73 \( 1 + (493. - 854. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-549. - 952. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.23e3T + 5.71e5T^{2} \)
89 \( 1 + (-355. - 616. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 636.T + 9.12e5T^{2} \)
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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

−11.40958682001698226507650158720, −10.71486854119476024537825175909, −9.956659430112666513678599754956, −8.937802242970339159441906694366, −8.111241156588033425936764931805, −6.73386210209998718630150022504, −5.28031068881158679689840501675, −4.03481265060090878645738344254, −2.51842679725669472074693620065, −1.24314492334852843060764835388, 0.45904137777442448573587882065, 2.77397451691573285216857958166, 4.37633257149682906748906521670, 5.92258138109053582850364619745, 6.89064209817416364071415111780, 7.39180156193479687934525488315, 8.627548106619484725052438402662, 9.418802326411353822662215701232, 10.42670451498194534606124737662, 11.80727436536537229594848204615

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