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Properties

Label 2-325-13.2-c2-0-16
Degree $2$
Conductor $325$
Sign $-0.374 - 0.927i$
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  + (2.97 + 0.797i)2-s + (2.45 + 4.25i)3-s + (4.74 + 2.74i)4-s + (3.91 + 14.6i)6-s + (−8.35 + 2.23i)7-s + (3.23 + 3.23i)8-s + (−7.58 + 13.1i)9-s + (3.15 − 11.7i)11-s + 26.9i·12-s + (10.4 + 7.74i)13-s − 26.6·14-s + (−3.93 − 6.81i)16-s + (22.2 + 12.8i)17-s + (−33.0 + 33.0i)18-s + (3.16 + 11.8i)19-s + ⋯
L(s)  = 1  + (1.48 + 0.398i)2-s + (0.819 + 1.41i)3-s + (1.18 + 0.685i)4-s + (0.652 + 2.43i)6-s + (−1.19 + 0.319i)7-s + (0.403 + 0.403i)8-s + (−0.842 + 1.45i)9-s + (0.287 − 1.07i)11-s + 2.24i·12-s + (0.803 + 0.595i)13-s − 1.90·14-s + (−0.245 − 0.425i)16-s + (1.30 + 0.754i)17-s + (−1.83 + 1.83i)18-s + (0.166 + 0.622i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.374 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.37870 + 3.52494i\)
\(L(\frac12)\) \(\approx\) \(2.37870 + 3.52494i\)
\(L(2)\) 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 \( 1 + (-10.4 - 7.74i)T \)
good2 \( 1 + (-2.97 - 0.797i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-2.45 - 4.25i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (8.35 - 2.23i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-3.15 + 11.7i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-22.2 - 12.8i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.16 - 11.8i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-4.18 + 2.41i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (8.72 + 15.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-37.9 + 37.9i)T - 961iT^{2} \)
37 \( 1 + (-13.5 + 50.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (20.0 + 5.38i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (26.4 + 15.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.7 - 52.7i)T + 2.20e3iT^{2} \)
53 \( 1 + 68.5T + 2.80e3T^{2} \)
59 \( 1 + (113. - 30.4i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (18.0 - 31.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (4.39 + 1.17i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (0.404 + 1.51i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (7.65 + 7.65i)T + 5.32e3iT^{2} \)
79 \( 1 - 3.21T + 6.24e3T^{2} \)
83 \( 1 + (-40.0 + 40.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (3.19 - 11.9i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-2.18 - 8.16i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.88030129053944544186802634672, −10.76849823296199688350915030685, −9.710549068281261406075383787215, −9.045979531178259594753702480676, −7.895615480766359315804147251789, −6.18146787216724621686977091991, −5.78830317825771130556121267450, −4.29709889707700080651643997758, −3.59471282085608004749519896506, −2.96488580777446072983066063859, 1.34425976202437538473008559439, 2.90522703469456992677851865808, 3.37515205749641830890010663941, 4.98121323156458636168481647022, 6.36414415914840296209144520263, 6.88415371672814249207440936280, 7.965597157788561615675819105754, 9.217825495284554830944778744543, 10.31797126709755322952451788000, 11.76785382726724607272709901343

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