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Properties

Label 2-325-65.63-c1-0-15
Degree $2$
Conductor $325$
Sign $0.683 + 0.730i$
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  + (0.616 + 1.06i)2-s + (0.608 − 2.26i)3-s + (0.240 − 0.416i)4-s + (2.79 − 0.749i)6-s + (−2.09 − 1.20i)7-s + 3.05·8-s + (−2.18 − 1.25i)9-s + (0.721 + 0.193i)11-s + (−0.798 − 0.798i)12-s + (−3.31 − 1.41i)13-s − 2.98i·14-s + (1.40 + 2.43i)16-s + (4.58 − 1.22i)17-s − 3.10i·18-s + (1.17 + 4.37i)19-s + ⋯
L(s)  = 1  + (0.435 + 0.754i)2-s + (0.351 − 1.31i)3-s + (0.120 − 0.208i)4-s + (1.14 − 0.305i)6-s + (−0.791 − 0.456i)7-s + 1.08·8-s + (−0.727 − 0.419i)9-s + (0.217 + 0.0582i)11-s + (−0.230 − 0.230i)12-s + (−0.919 − 0.393i)13-s − 0.796i·14-s + (0.350 + 0.607i)16-s + (1.11 − 0.297i)17-s − 0.732i·18-s + (0.269 + 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71147 - 0.742640i\)
\(L(\frac12)\) \(\approx\) \(1.71147 - 0.742640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 + 1.41i)T \)
good2 \( 1 + (-0.616 - 1.06i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.608 + 2.26i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.09 + 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.721 - 0.193i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.58 + 1.22i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.17 - 4.37i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.11 + 1.90i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.31 + 2.49i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.01 - 6.01i)T + 31iT^{2} \)
37 \( 1 + (-0.517 + 0.298i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.40 - 5.25i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.05 - 11.4i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 11.2iT - 47T^{2} \)
53 \( 1 + (-0.469 - 0.469i)T + 53iT^{2} \)
59 \( 1 + (9.49 - 2.54i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.729 - 1.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.94 + 3.37i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.47 - 0.664i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 4.94T + 73T^{2} \)
79 \( 1 + 10.7iT - 79T^{2} \)
83 \( 1 - 1.96iT - 83T^{2} \)
89 \( 1 + (-3.40 + 12.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.87 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.97197529342173265615522576988, −10.32219903852919933284647963686, −9.787222636665235070810641716439, −7.932245720191117242816647912618, −7.67757759062923081748946103695, −6.54203072419404351178672620197, −6.04016250271046119559532031486, −4.58767051350886083632695169344, −2.90494571872970612883833346137, −1.30026125769291526040394912719, 2.45475913559497639711182304963, 3.45460902747609275611744464971, 4.27988042138942109334776405795, 5.39377124694625196453778902525, 6.89822034618508348808398884075, 8.143084186182828654691830628240, 9.346067868898658426846305312046, 9.953596997796844788783277116328, 10.67330099182181868285659110795, 11.98341899966405603420322932010

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