Properties

Label 2-245-245.208-c1-0-11
Degree $2$
Conductor $245$
Sign $0.956 - 0.291i$
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  + (−0.514 + 2.71i)2-s + (0.0531 − 1.42i)3-s + (−5.25 − 2.06i)4-s + (−2.17 − 0.504i)5-s + (3.83 + 0.875i)6-s + (2.44 − 1.01i)7-s + (5.37 − 8.54i)8-s + (0.974 + 0.0730i)9-s + (2.49 − 5.66i)10-s + (−0.0463 − 0.618i)11-s + (−3.21 + 7.36i)12-s + (0.334 − 0.116i)13-s + (1.50 + 7.16i)14-s + (−0.832 + 3.06i)15-s + (12.1 + 11.3i)16-s + (−1.29 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.363 + 1.92i)2-s + (0.0307 − 0.820i)3-s + (−2.62 − 1.03i)4-s + (−0.974 − 0.225i)5-s + (1.56 + 0.357i)6-s + (0.923 − 0.383i)7-s + (1.89 − 3.02i)8-s + (0.324 + 0.0243i)9-s + (0.787 − 1.79i)10-s + (−0.0139 − 0.186i)11-s + (−0.927 + 2.12i)12-s + (0.0927 − 0.0324i)13-s + (0.402 + 1.91i)14-s + (−0.214 + 0.792i)15-s + (3.04 + 2.82i)16-s + (−0.314 − 0.425i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.956 - 0.291i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773656 + 0.115239i\)
\(L(\frac12)\) \(\approx\) \(0.773656 + 0.115239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.17 + 0.504i)T \)
7 \( 1 + (-2.44 + 1.01i)T \)
good2 \( 1 + (0.514 - 2.71i)T + (-1.86 - 0.730i)T^{2} \)
3 \( 1 + (-0.0531 + 1.42i)T + (-2.99 - 0.224i)T^{2} \)
11 \( 1 + (0.0463 + 0.618i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-0.334 + 0.116i)T + (10.1 - 8.10i)T^{2} \)
17 \( 1 + (1.29 + 1.75i)T + (-5.01 + 16.2i)T^{2} \)
19 \( 1 + (-3.50 + 6.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.29 + 3.90i)T + (6.77 + 21.9i)T^{2} \)
29 \( 1 + (-0.477 - 0.380i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-3.02 + 1.74i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.631 - 0.275i)T + (25.1 + 27.1i)T^{2} \)
41 \( 1 + (2.54 - 0.579i)T + (36.9 - 17.7i)T^{2} \)
43 \( 1 + (6.37 - 4.00i)T + (18.6 - 38.7i)T^{2} \)
47 \( 1 + (-4.39 - 0.832i)T + (43.7 + 17.1i)T^{2} \)
53 \( 1 + (8.85 - 3.86i)T + (36.0 - 38.8i)T^{2} \)
59 \( 1 + (-6.11 + 1.88i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.75 - 0.688i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (3.93 + 1.05i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.82 - 12.3i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-7.05 + 1.33i)T + (67.9 - 26.6i)T^{2} \)
79 \( 1 + (-6.35 - 3.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.113 + 0.324i)T + (-64.8 - 51.7i)T^{2} \)
89 \( 1 + (0.604 - 8.06i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (10.6 + 10.6i)T + 97iT^{2} \)
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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

−12.43536437225461701410682674385, −11.18724677128280514283341419148, −9.803109030696379468276499571487, −8.568632879238651702260826659139, −7.967065094825235736347378758599, −7.26135322012554037746555444033, −6.53426440914981354960902128351, −5.02948498612205728778898367746, −4.26892576469214899264547392453, −0.808678918880526564443468100001, 1.71673350269761319333919453035, 3.43616536524036685665833269067, 4.14777134251910980502400474535, 5.12898268016602076167365197886, 7.76865638036622207886162663294, 8.436996440731501372302104545446, 9.557091918480785241782902996323, 10.32519913547847505277191621999, 11.05012323765542703645695199728, 11.92237563770635793529307357945

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