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Properties

Label 2-325-13.11-c2-0-34
Degree $2$
Conductor $325$
Sign $-0.990 - 0.137i$
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  + (0.581 − 2.17i)2-s + (−1.25 − 2.17i)3-s + (−0.912 − 0.526i)4-s + (−5.44 + 1.45i)6-s + (−2.82 − 10.5i)7-s + (4.68 − 4.68i)8-s + (1.36 − 2.35i)9-s + (18.8 + 5.05i)11-s + 2.64i·12-s + (−12.7 − 2.28i)13-s − 24.5·14-s + (−9.55 − 16.5i)16-s + (16.9 + 9.77i)17-s + (−4.32 − 4.32i)18-s + (−24.2 + 6.50i)19-s + ⋯
L(s)  = 1  + (0.290 − 1.08i)2-s + (−0.417 − 0.723i)3-s + (−0.228 − 0.131i)4-s + (−0.906 + 0.243i)6-s + (−0.403 − 1.50i)7-s + (0.585 − 0.585i)8-s + (0.151 − 0.261i)9-s + (1.71 + 0.459i)11-s + 0.220i·12-s + (−0.984 − 0.175i)13-s − 1.75·14-s + (−0.597 − 1.03i)16-s + (0.995 + 0.574i)17-s + (−0.240 − 0.240i)18-s + (−1.27 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.990 - 0.137i$
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: \((2,\ 325,\ (\ :1),\ -0.990 - 0.137i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.117990 + 1.70766i\)
\(L(\frac12)\) \(\approx\) \(0.117990 + 1.70766i\)
\(L(2)\) 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 \( 1 + (12.7 + 2.28i)T \)
good2 \( 1 + (-0.581 + 2.17i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (1.25 + 2.17i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (2.82 + 10.5i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-18.8 - 5.05i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-16.9 - 9.77i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (24.2 - 6.50i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (22.2 - 12.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.96 + 6.85i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (6.84 + 6.84i)T + 961iT^{2} \)
37 \( 1 + (-38.1 - 10.2i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-8.85 + 33.0i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-21.1 - 12.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-6.86 + 6.86i)T - 2.20e3iT^{2} \)
53 \( 1 - 15.2T + 2.80e3T^{2} \)
59 \( 1 + (-9.62 - 35.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.6 - 110. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-118. + 31.8i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-73.2 + 73.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 16.9T + 6.24e3T^{2} \)
83 \( 1 + (64.8 + 64.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (-83.4 - 22.3i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (18.0 - 4.82i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.08437180052289690962152316180, −10.08049798780738156794831004175, −9.603064015940355762457886309613, −7.70370759892402137375446308459, −7.01102514640632065738013784864, −6.20452382244812270672677995939, −4.17032545286138538264162980961, −3.75881295772670484041843542127, −1.85741218616513071593053415974, −0.798885644863762940112243441092, 2.27228753005707098905587932100, 4.13184077357042683072221696623, 5.16401399669385563893347924615, 5.99113234235695124410398165314, 6.72114911629358638629694553775, 8.008360300572556252425329520320, 9.067392271770912859243163585986, 9.806400729766390353753179600440, 11.04551175540393169635826641753, 11.85755294805833266872092164701

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