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Properties

Label 2-245-245.202-c1-0-12
Degree $2$
Conductor $245$
Sign $-0.374 + 0.927i$
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  + (−2.14 + 0.242i)2-s + (−0.397 − 1.13i)3-s + (2.61 − 0.596i)4-s + (−2.20 + 0.348i)5-s + (1.13 + 2.34i)6-s + (1.57 + 2.12i)7-s + (−1.38 + 0.485i)8-s + (1.21 − 0.965i)9-s + (4.66 − 1.28i)10-s + (1.87 − 2.34i)11-s + (−1.71 − 2.73i)12-s + (−4.16 + 0.469i)13-s + (−3.90 − 4.18i)14-s + (1.27 + 2.37i)15-s + (−1.96 + 0.945i)16-s + (−3.66 − 5.82i)17-s + ⋯
L(s)  = 1  + (−1.51 + 0.171i)2-s + (−0.229 − 0.656i)3-s + (1.30 − 0.298i)4-s + (−0.987 + 0.155i)5-s + (0.461 + 0.958i)6-s + (0.596 + 0.802i)7-s + (−0.490 + 0.171i)8-s + (0.403 − 0.321i)9-s + (1.47 − 0.405i)10-s + (0.564 − 0.708i)11-s + (−0.495 − 0.788i)12-s + (−1.15 + 0.130i)13-s + (−1.04 − 1.11i)14-s + (0.329 + 0.612i)15-s + (−0.491 + 0.236i)16-s + (−0.888 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.374 + 0.927i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190168 - 0.281938i\)
\(L(\frac12)\) \(\approx\) \(0.190168 - 0.281938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.20 - 0.348i)T \)
7 \( 1 + (-1.57 - 2.12i)T \)
good2 \( 1 + (2.14 - 0.242i)T + (1.94 - 0.445i)T^{2} \)
3 \( 1 + (0.397 + 1.13i)T + (-2.34 + 1.87i)T^{2} \)
11 \( 1 + (-1.87 + 2.34i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (4.16 - 0.469i)T + (12.6 - 2.89i)T^{2} \)
17 \( 1 + (3.66 + 5.82i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 + (3.48 + 2.18i)T + (9.97 + 20.7i)T^{2} \)
29 \( 1 + (6.24 + 1.42i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (-3.58 + 2.25i)T + (16.0 - 33.3i)T^{2} \)
41 \( 1 + (-1.86 + 3.87i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-1.36 + 3.91i)T + (-33.6 - 26.8i)T^{2} \)
47 \( 1 + (-0.635 - 5.64i)T + (-45.8 + 10.4i)T^{2} \)
53 \( 1 + (-9.88 - 6.21i)T + (22.9 + 47.7i)T^{2} \)
59 \( 1 + (7.55 - 3.63i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (0.749 + 0.171i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (9.24 + 9.24i)T + 67iT^{2} \)
71 \( 1 + (0.144 + 0.634i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.847 + 7.52i)T + (-71.1 - 16.2i)T^{2} \)
79 \( 1 + 5.69iT - 79T^{2} \)
83 \( 1 + (0.143 - 1.27i)T + (-80.9 - 18.4i)T^{2} \)
89 \( 1 + (-5.50 - 6.90i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-8.11 + 8.11i)T - 97iT^{2} \)
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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

−11.69742467231915712660795722195, −10.93263013172585937071883435065, −9.482440282550022508901912254564, −8.948375017206566573203475801439, −7.64227460610719650710805215952, −7.39162534723706139238684781452, −6.14354954215207461385770126708, −4.39465828660151952852032437670, −2.28465052075576031642845015965, −0.46861524992269290499513775702, 1.65744978994552394552320958401, 4.01142270307855929632592620062, 4.83554149728130100940510472022, 7.05978216295954425306750992297, 7.60968841104500945410245251153, 8.543796109587103157178132786803, 9.663700674520171369494271148572, 10.35734369336455460590895386321, 11.08862039734009745386675492736, 11.88224437940083472969296947107

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