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Properties

Label 2-245-35.3-c1-0-11
Degree $2$
Conductor $245$
Sign $0.470 + 0.882i$
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  + (0.5 + 0.133i)2-s + (0.133 + 0.5i)3-s + (−1.5 − 0.866i)4-s + (−0.133 − 2.23i)5-s + 0.267i·6-s + (−1.36 − 1.36i)8-s + (2.36 − 1.36i)9-s + (0.232 − 1.13i)10-s + (1.36 − 2.36i)11-s + (0.232 − 0.866i)12-s + (2 − 2i)13-s + (1.09 − 0.366i)15-s + (1.23 + 2.13i)16-s + (−3.73 + i)17-s + (1.36 − 0.366i)18-s + (−0.366 − 0.633i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.0947i)2-s + (0.0773 + 0.288i)3-s + (−0.750 − 0.433i)4-s + (−0.0599 − 0.998i)5-s + 0.109i·6-s + (−0.482 − 0.482i)8-s + (0.788 − 0.455i)9-s + (0.0733 − 0.358i)10-s + (0.411 − 0.713i)11-s + (0.0669 − 0.249i)12-s + (0.554 − 0.554i)13-s + (0.283 − 0.0945i)15-s + (0.308 + 0.533i)16-s + (−0.905 + 0.242i)17-s + (0.321 − 0.0862i)18-s + (−0.0839 − 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08962 - 0.653593i\)
\(L(\frac12)\) \(\approx\) \(1.08962 - 0.653593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.133i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-0.133 - 0.5i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (3.73 - i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.366 + 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0358 + 0.133i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-6.46 - 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.73 + 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 - 2.83i)T + 43iT^{2} \)
47 \( 1 + (2.36 - 8.83i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.09 + 7.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.33 - 0.767i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.86 - 10.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (-3.46 - 12.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.83 + 1.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.09 + 2.09i)T - 83iT^{2} \)
89 \( 1 + (-0.330 - 0.571i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.92 - 5.92i)T + 97iT^{2} \)
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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

−12.23104889938478592756861137384, −10.88917532513824123228661787428, −9.869105156586312128795278780396, −8.954731618563697122362718758000, −8.402639803925009913688642744080, −6.66197633749224826987154892839, −5.56843992951423429966437739872, −4.52847395552872949655662830893, −3.67391235720223874429829754519, −1.03753174663562713164984530870, 2.26629895215786835342617923308, 3.80868590614437506270269041293, 4.68611151283809022353215390164, 6.34430712246511704204514345142, 7.23880411338807983794650056660, 8.257376510738481310199078478030, 9.424548921185540525125873274086, 10.33611423026583847531680027826, 11.50971000050284844837755475437, 12.25979848950425048614559795821

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