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Properties

Label 2-245-35.2-c2-0-33
Degree $2$
Conductor $245$
Sign $0.847 + 0.530i$
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  + (2.95 − 0.792i)2-s + (2.36 + 0.633i)3-s + (4.66 − 2.69i)4-s + (4.16 − 2.76i)5-s + 7.49·6-s + (2.99 − 2.99i)8-s + (−2.60 − 1.50i)9-s + (10.1 − 11.4i)10-s + (−3.12 − 5.40i)11-s + (12.7 − 3.40i)12-s + (−11.7 + 11.7i)13-s + (11.6 − 3.88i)15-s + (−4.28 + 7.41i)16-s + (−1.22 + 4.56i)17-s + (−8.89 − 2.38i)18-s + (11.4 + 6.58i)19-s + ⋯
L(s)  = 1  + (1.47 − 0.396i)2-s + (0.788 + 0.211i)3-s + (1.16 − 0.672i)4-s + (0.833 − 0.552i)5-s + 1.24·6-s + (0.373 − 0.373i)8-s + (−0.289 − 0.167i)9-s + (1.01 − 1.14i)10-s + (−0.283 − 0.491i)11-s + (1.06 − 0.284i)12-s + (−0.902 + 0.902i)13-s + (0.773 − 0.259i)15-s + (−0.267 + 0.463i)16-s + (−0.0720 + 0.268i)17-s + (−0.494 − 0.132i)18-s + (0.600 + 0.346i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.847 + 0.530i$
Analytic conductor: \(6.67576\)
Root analytic conductor: \(2.58375\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1),\ 0.847 + 0.530i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.21924 - 1.21065i\)
\(L(\frac12)\) \(\approx\) \(4.21924 - 1.21065i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.16 + 2.76i)T \)
7 \( 1 \)
good2 \( 1 + (-2.95 + 0.792i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (-2.36 - 0.633i)T + (7.79 + 4.5i)T^{2} \)
11 \( 1 + (3.12 + 5.40i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (11.7 - 11.7i)T - 169iT^{2} \)
17 \( 1 + (1.22 - 4.56i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-11.4 - 6.58i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.33 - 34.8i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + 28.5iT - 841T^{2} \)
31 \( 1 + (10.1 + 17.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-22.6 + 6.06i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 45.1T + 1.68e3T^{2} \)
43 \( 1 + (21.9 - 21.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-10.4 + 2.79i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (71.6 + 19.2i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-14.6 + 8.43i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (16.7 - 29.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-25.6 + 95.8i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (99.8 + 26.7i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (36.1 + 20.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (7.39 - 7.39i)T - 6.88e3iT^{2} \)
89 \( 1 + (-19.2 - 11.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
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

−11.96997960615035308637407148142, −11.22081436413852777768477405827, −9.726873960793209191051288661423, −9.133344544444085092606617575750, −7.84194161677066915980293181812, −6.21235587265488479381619382685, −5.40279859886606986113841074328, −4.29214122341033174945256403066, −3.11465096495141256959580777424, −2.02637363893903790977091184002, 2.49707981304314763093483294545, 3.09091247216259254213893709940, 4.81322524193665652022555447671, 5.60712626026390174944956268798, 6.83087764238423928415532957554, 7.59222446481209552121205825707, 8.999234096004190736852072448850, 10.08195117906956137017808764186, 11.17436596161630728201956260044, 12.56223955224426158679061727751

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