Properties

Label 2-325-13.3-c1-0-13
Degree $2$
Conductor $325$
Sign $0.859 + 0.511i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (0.809 + 1.40i)2-s + (−1.11 − 1.93i)3-s + (−0.309 + 0.535i)4-s + (1.80 − 3.13i)6-s + (−0.118 + 0.204i)7-s + 2.23·8-s + (−1 + 1.73i)9-s + (−2.11 − 3.66i)11-s + 1.38·12-s + (1 − 3.46i)13-s − 0.381·14-s + (2.42 + 4.20i)16-s + (2.73 − 4.73i)17-s − 3.23·18-s + (0.118 − 0.204i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.990i)2-s + (−0.645 − 1.11i)3-s + (−0.154 + 0.267i)4-s + (0.738 − 1.27i)6-s + (−0.0446 + 0.0772i)7-s + 0.790·8-s + (−0.333 + 0.577i)9-s + (−0.638 − 1.10i)11-s + 0.398·12-s + (0.277 − 0.960i)13-s − 0.102·14-s + (0.606 + 1.05i)16-s + (0.663 − 1.14i)17-s − 0.762·18-s + (0.0270 − 0.0469i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44859 - 0.398123i\)
\(L(\frac12)\) \(\approx\) \(1.44859 - 0.398123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + (-0.809 - 1.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.118 + 0.204i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.97 - 5.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.881 - 1.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (6.35 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.20 + 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.35 - 9.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.881 + 1.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.73 + 4.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.55828148208471593298850415137, −10.95412553674207765490254482214, −9.646197803898578719857196614600, −8.040107285992264161983795794839, −7.55347861868168264227253097549, −6.53467463696251984531624811190, −5.71608149209922709986782640188, −5.12048435712763670274353868291, −3.17444443030196359609939706780, −1.07479646069549256135632957733, 2.03214319651505842861828524135, 3.60236814137968309890456470238, 4.47698604952670919961962973509, 5.18610495781063016831296549462, 6.66629651106512238511901918883, 7.946667428119654024854326274085, 9.366455143692521586295709792065, 10.31674451286190025880640002540, 10.70499298177655336098048784137, 11.57763077785528128510305960317

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