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Properties

Label 2-260-65.2-c1-0-2
Degree $2$
Conductor $260$
Sign $0.966 - 0.257i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
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  + (0.834 + 0.223i)3-s + (2.20 + 0.382i)5-s + (2.07 + 1.19i)7-s + (−1.95 − 1.12i)9-s + (0.0379 − 0.141i)11-s + (−3.58 − 0.344i)13-s + (1.75 + 0.811i)15-s + (1.33 + 4.96i)17-s + (4.18 − 1.12i)19-s + (1.46 + 1.46i)21-s + (2.28 − 8.53i)23-s + (4.70 + 1.68i)25-s + (−3.20 − 3.20i)27-s + (−5.00 + 2.88i)29-s + (−4.94 + 4.94i)31-s + ⋯
L(s)  = 1  + (0.481 + 0.129i)3-s + (0.985 + 0.171i)5-s + (0.783 + 0.452i)7-s + (−0.650 − 0.375i)9-s + (0.0114 − 0.0427i)11-s + (−0.995 − 0.0956i)13-s + (0.452 + 0.209i)15-s + (0.322 + 1.20i)17-s + (0.960 − 0.257i)19-s + (0.318 + 0.318i)21-s + (0.476 − 1.77i)23-s + (0.941 + 0.337i)25-s + (−0.617 − 0.617i)27-s + (−0.928 + 0.536i)29-s + (−0.887 + 0.887i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65901 + 0.216875i\)
\(L(\frac12)\) \(\approx\) \(1.65901 + 0.216875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.20 - 0.382i)T \)
13 \( 1 + (3.58 + 0.344i)T \)
good3 \( 1 + (-0.834 - 0.223i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.07 - 1.19i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0379 + 0.141i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.33 - 4.96i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.18 + 1.12i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.28 + 8.53i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.00 - 2.88i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.94 - 4.94i)T - 31iT^{2} \)
37 \( 1 + (-3.76 + 2.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (11.6 + 3.11i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.642 + 0.172i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 3.88iT - 47T^{2} \)
53 \( 1 + (2.38 - 2.38i)T - 53iT^{2} \)
59 \( 1 + (1.94 + 7.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.54 - 9.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.92 + 3.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.66 - 6.21i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 0.839T + 73T^{2} \)
79 \( 1 - 3.64iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (11.7 + 3.15i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.254 - 0.440i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.10228358431490362711725592629, −10.97501562240581417121484647541, −10.07125472794795383000649218430, −9.052390961733861302335249311302, −8.402490847729018072918146213671, −7.08766911659156786977509287736, −5.83248289192419170343278344293, −4.96391913670128915214920195195, −3.19333844098892573125057195413, −1.98335022780897355120309935801, 1.74092905913134573681512071315, 3.07787648255529334789417087150, 4.92644184011180749827072350589, 5.62263108587678815958422664357, 7.29542398296079552907373077486, 7.893212286791023322509579761117, 9.328970851904908789933937200543, 9.717600318725400754508813621719, 11.12262686452780623320502702922, 11.79543344232184062846697388886

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