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Properties

Label 2-245-245.143-c1-0-12
Degree $2$
Conductor $245$
Sign $0.926 - 0.375i$
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  + (−1.07 + 0.203i)2-s + (3.30 − 0.123i)3-s + (−0.740 + 0.290i)4-s + (1.80 + 1.32i)5-s + (−3.53 + 0.806i)6-s + (−2.12 − 1.58i)7-s + (2.59 − 1.63i)8-s + (7.88 − 0.591i)9-s + (−2.21 − 1.05i)10-s + (−0.0263 + 0.352i)11-s + (−2.40 + 1.05i)12-s + (−1.19 + 3.40i)13-s + (2.60 + 1.27i)14-s + (6.11 + 4.13i)15-s + (−1.30 + 1.20i)16-s + (−0.353 − 0.260i)17-s + ⋯
L(s)  = 1  + (−0.762 + 0.144i)2-s + (1.90 − 0.0713i)3-s + (−0.370 + 0.145i)4-s + (0.807 + 0.590i)5-s + (−1.44 + 0.329i)6-s + (−0.801 − 0.598i)7-s + (0.918 − 0.577i)8-s + (2.62 − 0.197i)9-s + (−0.700 − 0.333i)10-s + (−0.00795 + 0.106i)11-s + (−0.695 + 0.303i)12-s + (−0.330 + 0.945i)13-s + (0.697 + 0.340i)14-s + (1.57 + 1.06i)15-s + (−0.325 + 0.301i)16-s + (−0.0856 − 0.0632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43322 + 0.279516i\)
\(L(\frac12)\) \(\approx\) \(1.43322 + 0.279516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.80 - 1.32i)T \)
7 \( 1 + (2.12 + 1.58i)T \)
good2 \( 1 + (1.07 - 0.203i)T + (1.86 - 0.730i)T^{2} \)
3 \( 1 + (-3.30 + 0.123i)T + (2.99 - 0.224i)T^{2} \)
11 \( 1 + (0.0263 - 0.352i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (1.19 - 3.40i)T + (-10.1 - 8.10i)T^{2} \)
17 \( 1 + (0.353 + 0.260i)T + (5.01 + 16.2i)T^{2} \)
19 \( 1 + (1.85 + 3.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.31 + 3.13i)T + (-6.77 + 21.9i)T^{2} \)
29 \( 1 + (5.02 - 4.01i)T + (6.45 - 28.2i)T^{2} \)
31 \( 1 + (-5.37 - 3.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.63 + 3.75i)T + (-25.1 + 27.1i)T^{2} \)
41 \( 1 + (5.49 + 1.25i)T + (36.9 + 17.7i)T^{2} \)
43 \( 1 + (-3.12 + 4.97i)T + (-18.6 - 38.7i)T^{2} \)
47 \( 1 + (2.07 + 10.9i)T + (-43.7 + 17.1i)T^{2} \)
53 \( 1 + (-0.512 + 1.17i)T + (-36.0 - 38.8i)T^{2} \)
59 \( 1 + (6.01 + 1.85i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (6.61 + 2.59i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (0.0905 + 0.338i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.58 - 9.51i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.409 - 2.16i)T + (-67.9 - 26.6i)T^{2} \)
79 \( 1 + (11.4 - 6.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.36 + 1.52i)T + (64.8 - 51.7i)T^{2} \)
89 \( 1 + (1.01 + 13.5i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-4.04 - 4.04i)T + 97iT^{2} \)
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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

−12.61348574743102134440738518056, −10.49767252045402852573666244035, −9.889367078361240228833383833337, −9.165972555859318095408644427808, −8.558201347821142411175347354627, −7.21996734724885269931616578298, −6.85703905878062811703304401573, −4.38658639902194046163186895588, −3.29927815235256293665214506430, −1.96708902691646729486124649774, 1.74236468452146190831101539131, 2.96431220481101468514436807481, 4.44985555458667331415689978597, 5.95178575483758579696190738553, 7.74507412989788397582046121751, 8.351425357944831469007987265501, 9.248795408521971837576922086798, 9.712512492280314421252859734558, 10.33706588232536099935124555349, 12.41301469308001057147068042871

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