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Properties

Label 2-325-13.10-c1-0-7
Degree $2$
Conductor $325$
Sign $-0.770 - 0.637i$
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  + (−0.769 + 0.443i)2-s + (1.53 + 2.65i)3-s + (−0.605 + 1.04i)4-s + (−2.35 − 1.35i)6-s + (3.08 + 1.78i)7-s − 2.85i·8-s + (−3.18 + 5.52i)9-s + (0.816 − 0.471i)11-s − 3.70·12-s + (3.56 + 0.555i)13-s − 3.16·14-s + (0.0547 + 0.0947i)16-s + (0.479 − 0.831i)17-s − 5.66i·18-s + (−2.89 − 1.66i)19-s + ⋯
L(s)  = 1  + (−0.543 + 0.313i)2-s + (0.883 + 1.53i)3-s + (−0.302 + 0.524i)4-s + (−0.961 − 0.555i)6-s + (1.16 + 0.673i)7-s − 1.00i·8-s + (−1.06 + 1.84i)9-s + (0.246 − 0.142i)11-s − 1.07·12-s + (0.988 + 0.154i)13-s − 0.845·14-s + (0.0136 + 0.0236i)16-s + (0.116 − 0.201i)17-s − 1.33i·18-s + (−0.663 − 0.383i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452869 + 1.25787i\)
\(L(\frac12)\) \(\approx\) \(0.452869 + 1.25787i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.56 - 0.555i)T \)
good2 \( 1 + (0.769 - 0.443i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.53 - 2.65i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.08 - 1.78i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.816 + 0.471i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.479 + 0.831i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.89 + 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.29 + 5.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.19 + 7.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.13iT - 31T^{2} \)
37 \( 1 + (-6.90 + 3.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.690 + 0.398i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.50 - 7.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + (-0.869 - 0.501i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.26 - 3.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.969 + 0.559i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.60 + 0.926i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.69iT - 73T^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 + 0.187iT - 83T^{2} \)
89 \( 1 + (-8.21 + 4.74i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.3 + 8.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.62676117876368931169319718347, −10.88738532864818981310405167829, −9.759723553949895444850884050685, −9.055243232943958816493428129340, −8.370550006947966096357702586278, −7.86429457116142057964543475864, −6.02096677756106459738430323630, −4.53983697699398811035119143848, −3.99849205725450207059790136705, −2.54450745146126997411553642482, 1.24079452762513296889187362105, 1.90242884437752985549598934663, 3.74145617080233519846337213933, 5.42401549725175438120635296125, 6.65284459938909396922467748282, 7.74245051250704507166442616386, 8.353115353146438827921602078021, 9.060275706561948251034563738035, 10.35747369358668090722764671339, 11.24334780193942374188522181143

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