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Properties

Label 2-245-35.33-c1-0-7
Degree $2$
Conductor $245$
Sign $0.730 + 0.683i$
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  + (−0.366 − 1.36i)2-s + (2.15 + 0.578i)3-s + (0.578 + 2.15i)5-s − 3.16i·6-s + (−2 − 1.99i)8-s + (1.73 + i)9-s + (2.73 − 1.58i)10-s + (0.5 + 0.866i)11-s + (1.58 − 1.58i)13-s + 5i·15-s + (−1.99 + 3.46i)16-s + (−0.578 + 2.15i)17-s + (0.732 − 2.73i)18-s + (1.58 − 2.73i)19-s + (0.999 − i)22-s + (−2.73 + 0.732i)23-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (1.24 + 0.334i)3-s + (0.258 + 0.965i)5-s − 1.29i·6-s + (−0.707 − 0.707i)8-s + (0.577 + 0.333i)9-s + (0.866 − 0.499i)10-s + (0.150 + 0.261i)11-s + (0.438 − 0.438i)13-s + 1.29i·15-s + (−0.499 + 0.866i)16-s + (−0.140 + 0.523i)17-s + (0.172 − 0.643i)18-s + (0.362 − 0.628i)19-s + (0.213 − 0.213i)22-s + (−0.569 + 0.152i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.730 + 0.683i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.730 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58682 - 0.626450i\)
\(L(\frac12)\) \(\approx\) \(1.58682 - 0.626450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.578 - 2.15i)T \)
7 \( 1 \)
good2 \( 1 + (0.366 + 1.36i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-2.15 - 0.578i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 + 1.58i)T - 13iT^{2} \)
17 \( 1 + (0.578 - 2.15i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.58 + 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.73 - 0.732i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (2.73 - 1.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.19 + 8.19i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.48iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (6.47 - 1.73i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.74 - 8.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.47 - 3.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.2 - 6.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 3.16i)T - 83iT^{2} \)
89 \( 1 + (3.16 - 5.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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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

−11.70527825630383266938343668320, −10.88445970931639405225387475278, −10.02289049555213809724842984784, −9.402398126304978778450513995117, −8.346390779703778330615358067708, −7.13143424904601891871084149540, −5.96748001431216060540718327220, −3.85971675028422542627629638814, −3.01737037788262220371250875785, −2.01463159219255983548288525210, 1.96304829044931886536450323833, 3.49381406304248483237396304610, 5.18544837824832205713865112159, 6.36522198792718023490815995540, 7.49238570861021933360152618412, 8.354006253689432550925733168507, 8.833114605848637783696132659077, 9.746248760927549310608825602026, 11.43740569214696400832150975132, 12.36958995956158968705147962964

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