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Properties

Label 2-245-245.108-c1-0-0
Degree $2$
Conductor $245$
Sign $0.0141 + 0.999i$
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  + (−1.06 + 0.787i)2-s + (−0.505 + 2.66i)3-s + (−0.0709 + 0.230i)4-s + (−0.576 − 2.16i)5-s + (−1.56 − 3.24i)6-s + (−2.33 − 1.25i)7-s + (−0.981 − 2.80i)8-s + (−4.07 − 1.60i)9-s + (2.31 + 1.85i)10-s + (−0.869 − 2.21i)11-s + (−0.578 − 0.305i)12-s + (0.458 + 4.06i)13-s + (3.47 − 0.499i)14-s + (6.05 − 0.448i)15-s + (2.85 + 1.94i)16-s + (0.0973 + 2.60i)17-s + ⋯
L(s)  = 1  + (−0.754 + 0.556i)2-s + (−0.291 + 1.54i)3-s + (−0.0354 + 0.115i)4-s + (−0.257 − 0.966i)5-s + (−0.638 − 1.32i)6-s + (−0.880 − 0.473i)7-s + (−0.347 − 0.991i)8-s + (−1.35 − 0.533i)9-s + (0.732 + 0.585i)10-s + (−0.262 − 0.668i)11-s + (−0.166 − 0.0882i)12-s + (0.127 + 1.12i)13-s + (0.928 − 0.133i)14-s + (1.56 − 0.115i)15-s + (0.714 + 0.487i)16-s + (0.0236 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.0141 + 0.999i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.0141 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0109352 - 0.0107819i\)
\(L(\frac12)\) \(\approx\) \(0.0109352 - 0.0107819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.576 + 2.16i)T \)
7 \( 1 + (2.33 + 1.25i)T \)
good2 \( 1 + (1.06 - 0.787i)T + (0.589 - 1.91i)T^{2} \)
3 \( 1 + (0.505 - 2.66i)T + (-2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.869 + 2.21i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.458 - 4.06i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (-0.0973 - 2.60i)T + (-16.9 + 1.27i)T^{2} \)
19 \( 1 + (2.06 + 3.56i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.64 + 0.136i)T + (22.9 + 1.71i)T^{2} \)
29 \( 1 + (0.312 + 0.0712i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (5.82 + 3.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.64 - 5.01i)T + (-20.8 - 30.5i)T^{2} \)
41 \( 1 + (-4.47 + 9.29i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.83 + 3.44i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (-3.32 - 4.50i)T + (-13.8 + 44.9i)T^{2} \)
53 \( 1 + (5.47 + 10.3i)T + (-29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.105 - 1.41i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (-4.27 - 13.8i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.333 + 1.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.45 - 6.39i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-3.40 + 4.61i)T + (-21.5 - 69.7i)T^{2} \)
79 \( 1 + (5.92 - 3.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.39 + 0.720i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (5.24 - 13.3i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (-1.25 - 1.25i)T + 97iT^{2} \)
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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

−12.77713104193127754142419975346, −11.65684063185536526967423850472, −10.58000500063839666404745265416, −9.657031556878471260906462517823, −9.049137355804821081802288097317, −8.319889628378116313768514435673, −6.89643205501504997085015112891, −5.68728171489921561327040912641, −4.26571233452887499191340701629, −3.67053263725749897265475591114, 0.01545499336918777097090731991, 1.94310095459934504675516620096, 3.03916906031809104342496222101, 5.60361566213511306461490994626, 6.42082787647052699057251593940, 7.47761429399547600370464904916, 8.264932234414633110104131701799, 9.652419450464671147006679072860, 10.43701115785592371513053621641, 11.38497004326829096749542015454

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