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Properties

Label 2-300-100.23-c1-0-10
Degree $2$
Conductor $300$
Sign $0.997 - 0.0722i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−1.35 + 0.410i)2-s + (−0.453 − 0.891i)3-s + (1.66 − 1.11i)4-s + (2.17 + 0.536i)5-s + (0.980 + 1.01i)6-s + (0.403 + 0.403i)7-s + (−1.79 + 2.18i)8-s + (−0.587 + 0.809i)9-s + (−3.15 + 0.163i)10-s + (2.68 + 3.69i)11-s + (−1.74 − 0.977i)12-s + (−0.669 − 4.22i)13-s + (−0.711 − 0.380i)14-s + (−0.507 − 2.17i)15-s + (1.53 − 3.69i)16-s + (1.13 + 0.576i)17-s + ⋯
L(s)  = 1  + (−0.956 + 0.290i)2-s + (−0.262 − 0.514i)3-s + (0.831 − 0.555i)4-s + (0.970 + 0.240i)5-s + (0.400 + 0.416i)6-s + (0.152 + 0.152i)7-s + (−0.634 + 0.772i)8-s + (−0.195 + 0.269i)9-s + (−0.998 + 0.0518i)10-s + (0.809 + 1.11i)11-s + (−0.503 − 0.282i)12-s + (−0.185 − 1.17i)13-s + (−0.190 − 0.101i)14-s + (−0.130 − 0.562i)15-s + (0.383 − 0.923i)16-s + (0.274 + 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 - 0.0722i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.997 - 0.0722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.973615 + 0.0352290i\)
\(L(\frac12)\) \(\approx\) \(0.973615 + 0.0352290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.410i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (-2.17 - 0.536i)T \)
good7 \( 1 + (-0.403 - 0.403i)T + 7iT^{2} \)
11 \( 1 + (-2.68 - 3.69i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.669 + 4.22i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.13 - 0.576i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.214 - 0.658i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.103 + 0.654i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-5.48 - 1.78i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.87 + 2.55i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.551 - 0.0872i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.58 + 2.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.72 + 2.72i)T - 43iT^{2} \)
47 \( 1 + (9.06 - 4.61i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-7.66 + 3.90i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.72 + 4.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (10.7 - 7.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.20 - 8.25i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-1.71 - 0.555i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.95 + 1.10i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.36 + 7.28i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.97 + 4.06i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-3.05 - 4.19i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.74 - 11.2i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.73400826855597451118146048320, −10.41589560990159361181990181580, −10.01160753016765472460915705574, −8.924384382883089755374715707923, −7.890671170408193074041925400724, −6.88731724408967444053371412924, −6.12987797857883753627720792985, −5.07587861456820082776668161420, −2.67187069846946439233262975095, −1.39353715258667111376052832506, 1.31655361129911993235646310430, 2.98626802236733595028183898559, 4.51628836061471365496129586739, 6.05412984905341913354584547727, 6.73807286085474537855419435217, 8.298375400119274844979951850837, 9.102136602675691366874849149292, 9.766538672511312211772298629054, 10.65416768568734290537382810165, 11.55502795141173593764688950111

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