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Properties

Label 2-325-13.11-c2-0-29
Degree $2$
Conductor $325$
Sign $0.321 + 0.946i$
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  + (0.246 − 0.921i)2-s + (0.816 + 1.41i)3-s + (2.67 + 1.54i)4-s + (1.50 − 0.403i)6-s + (−3.13 − 11.6i)7-s + (4.78 − 4.78i)8-s + (3.16 − 5.48i)9-s + (−11.2 − 3.01i)11-s + 5.04i·12-s + (−3.70 − 12.4i)13-s − 11.5·14-s + (2.95 + 5.12i)16-s + (8.02 + 4.63i)17-s + (−4.27 − 4.27i)18-s + (4.00 − 1.07i)19-s + ⋯
L(s)  = 1  + (0.123 − 0.460i)2-s + (0.272 + 0.471i)3-s + (0.669 + 0.386i)4-s + (0.250 − 0.0671i)6-s + (−0.447 − 1.67i)7-s + (0.597 − 0.597i)8-s + (0.351 − 0.609i)9-s + (−1.02 − 0.273i)11-s + 0.420i·12-s + (−0.285 − 0.958i)13-s − 0.825·14-s + (0.184 + 0.320i)16-s + (0.472 + 0.272i)17-s + (−0.237 − 0.237i)18-s + (0.211 − 0.0565i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.321 + 0.946i$
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: \((2,\ 325,\ (\ :1),\ 0.321 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72060 - 1.23236i\)
\(L(\frac12)\) \(\approx\) \(1.72060 - 1.23236i\)
\(L(2)\) 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 \( 1 + (3.70 + 12.4i)T \)
good2 \( 1 + (-0.246 + 0.921i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.816 - 1.41i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (3.13 + 11.6i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (11.2 + 3.01i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-8.02 - 4.63i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.00 + 1.07i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-12.3 + 7.15i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-15.5 - 26.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-18.2 - 18.2i)T + 961iT^{2} \)
37 \( 1 + (68.8 + 18.4i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (12.8 - 47.8i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-38.5 - 22.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-8.77 + 8.77i)T - 2.20e3iT^{2} \)
53 \( 1 + 9.87T + 2.80e3T^{2} \)
59 \( 1 + (12.3 + 46.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-15.9 + 27.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.65 - 32.2i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-60.2 + 16.1i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-4.72 + 4.72i)T - 5.32e3iT^{2} \)
79 \( 1 - 37.3T + 6.24e3T^{2} \)
83 \( 1 + (5.54 + 5.54i)T + 6.88e3iT^{2} \)
89 \( 1 + (-125. - 33.6i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (110. - 29.6i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−10.77427708988468366796087352365, −10.51603861301680577911095768148, −9.762050036313300287565733015407, −8.231333760200855945351183944751, −7.34725080886687528040126850457, −6.57769866498683649066230309440, −4.88333343408983075020221241923, −3.61001274256579874192905904817, −3.04397851958084717129656181051, −0.949691625222218995712371075067, 1.98778239012729760452142467647, 2.70380822208052313432890843433, 4.95097610917595484463567555753, 5.69680299404410149324913894425, 6.78792257823497165616126251072, 7.60828074234105917335108216224, 8.575512845100276730359193305892, 9.690066809071250861213891373680, 10.61368789705240550896623278662, 11.83694148653230594189263113835

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