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Properties

Label 2-325-13.11-c2-0-28
Degree $2$
Conductor $325$
Sign $0.526 + 0.850i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.518 + 1.93i)2-s + (0.0786 + 0.136i)3-s + (−0.00647 − 0.00373i)4-s + (−0.304 + 0.0814i)6-s + (−2.27 − 8.48i)7-s + (−5.65 + 5.65i)8-s + (4.48 − 7.77i)9-s + (−9.01 − 2.41i)11-s − 0.00117i·12-s + (−6.41 − 11.3i)13-s + 17.5·14-s + (−8.01 − 13.8i)16-s + (−19.3 − 11.1i)17-s + (12.7 + 12.7i)18-s + (14.9 − 4.00i)19-s + ⋯
L(s)  = 1  + (−0.259 + 0.966i)2-s + (0.0262 + 0.0453i)3-s + (−0.00161 − 0.000933i)4-s + (−0.0506 + 0.0135i)6-s + (−0.324 − 1.21i)7-s + (−0.706 + 0.706i)8-s + (0.498 − 0.863i)9-s + (−0.819 − 0.219i)11-s − 9.78e − 5i·12-s + (−0.493 − 0.869i)13-s + 1.25·14-s + (−0.500 − 0.867i)16-s + (−1.13 − 0.657i)17-s + (0.705 + 0.705i)18-s + (0.786 − 0.210i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.526 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.715852 - 0.398793i\)
\(L(\frac12)\) \(\approx\) \(0.715852 - 0.398793i\)
\(L(2)\) 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 \( 1 + (6.41 + 11.3i)T \)
good2 \( 1 + (0.518 - 1.93i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.0786 - 0.136i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (2.27 + 8.48i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (9.01 + 2.41i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (19.3 + 11.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.9 + 4.00i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (31.2 - 18.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (22.1 + 38.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.1 - 14.1i)T + 961iT^{2} \)
37 \( 1 + (-54.1 - 14.5i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (2.71 - 10.1i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-12.8 - 7.41i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.7 + 36.7i)T - 2.20e3iT^{2} \)
53 \( 1 + 4.53T + 2.80e3T^{2} \)
59 \( 1 + (-4.91 - 18.3i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (42.0 - 72.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-13.5 + 50.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-107. + 28.7i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (13.6 - 13.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 30.3T + 6.24e3T^{2} \)
83 \( 1 + (-23.0 - 23.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (52.8 + 14.1i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (105. - 28.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.20900216933030857440488745918, −10.07568719348277015377764418018, −9.386016123631787826917380752460, −7.983836253575632783408185467987, −7.44421059766822586307370423890, −6.58506315850474305263799616564, −5.55140129357041984063976605402, −4.14269847434935883130158547170, −2.79119798705927973955828327168, −0.38497007746117929495916338168, 1.96927521752897157348255646929, 2.57559236085165049751047401690, 4.25317477377430969445471180650, 5.59540489079771752679628462731, 6.66065369111301657499989386892, 7.893747254308124535144920799265, 9.061851023621764757952064387976, 9.784227108276662528789897459621, 10.68416663169033060045836564678, 11.43652640271538006494159371352

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