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Properties

Label 2-925-185.23-c1-0-13
Degree $2$
Conductor $925$
Sign $0.958 + 0.284i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (−0.983 − 0.567i)2-s + (−1.02 − 0.273i)3-s + (−0.354 − 0.614i)4-s + (0.849 + 0.849i)6-s + (−1.57 − 0.422i)7-s + 3.07i·8-s + (−1.63 − 0.941i)9-s + 1.92i·11-s + (0.194 + 0.724i)12-s + (1.32 − 0.763i)13-s + (1.31 + 1.31i)14-s + (1.03 − 1.79i)16-s + (−0.660 + 1.14i)17-s + (1.06 + 1.85i)18-s + (−2.17 − 0.583i)19-s + ⋯
L(s)  = 1  + (−0.695 − 0.401i)2-s + (−0.589 − 0.157i)3-s + (−0.177 − 0.307i)4-s + (0.346 + 0.346i)6-s + (−0.596 − 0.159i)7-s + 1.08i·8-s + (−0.543 − 0.313i)9-s + 0.579i·11-s + (0.0560 + 0.209i)12-s + (0.366 − 0.211i)13-s + (0.350 + 0.350i)14-s + (0.259 − 0.449i)16-s + (−0.160 + 0.277i)17-s + (0.251 + 0.436i)18-s + (−0.499 − 0.133i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.958 + 0.284i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528921 - 0.0768272i\)
\(L(\frac12)\) \(\approx\) \(0.528921 - 0.0768272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (-5.83 - 1.71i)T \)
good2 \( 1 + (0.983 + 0.567i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.02 + 0.273i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.57 + 0.422i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 - 1.92iT - 11T^{2} \)
13 \( 1 + (-1.32 + 0.763i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.660 - 1.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.17 + 0.583i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.57iT - 23T^{2} \)
29 \( 1 + (0.241 + 0.241i)T + 29iT^{2} \)
31 \( 1 + (1.00 - 1.00i)T - 31iT^{2} \)
41 \( 1 + (-6.76 + 3.90i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 5.46iT - 43T^{2} \)
47 \( 1 + (1.79 - 1.79i)T - 47iT^{2} \)
53 \( 1 + (3.24 - 0.870i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.40 - 5.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-10.4 - 2.80i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.53 + 9.45i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.65 - 2.86i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.78 + 1.78i)T - 73iT^{2} \)
79 \( 1 + (-12.3 - 3.30i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-8.57 + 2.29i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-16.4 + 4.40i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−10.01814388998907091578250609078, −9.322131982977923993943146587943, −8.615103319707173011508916854375, −7.60745172511248494746709315354, −6.48674518353331059600085253368, −5.80483474953450110360632931452, −4.88230219147383461694291087385, −3.56938124072190653546311269082, −2.19954860688081108221840022576, −0.78868947553208777751941116135, 0.53119403915566800467313892850, 2.67473138936743247582225892661, 3.82504218133447302854115076077, 4.89004296284835766354356708904, 6.14561328741537955854071828206, 6.53375675076229366051911803063, 7.79355204023195833244180613999, 8.431742850896848586329500502296, 9.199352950060359291969815635876, 9.947545311134544322973450477622

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