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Properties

Label 2-750-125.91-c1-0-3
Degree $2$
Conductor $750$
Sign $-0.235 - 0.971i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (0.728 + 0.684i)2-s + (0.535 − 0.844i)3-s + (0.0627 + 0.998i)4-s + (−2.15 − 0.592i)5-s + (0.968 − 0.248i)6-s + (1.47 + 4.55i)7-s + (−0.637 + 0.770i)8-s + (−0.425 − 0.904i)9-s + (−1.16 − 1.90i)10-s + (−2.49 − 2.34i)11-s + (0.876 + 0.481i)12-s + (1.50 + 3.19i)13-s + (−2.03 + 4.33i)14-s + (−1.65 + 1.50i)15-s + (−0.992 + 0.125i)16-s + (−0.230 + 3.66i)17-s + ⋯
L(s)  = 1  + (0.515 + 0.484i)2-s + (0.309 − 0.487i)3-s + (0.0313 + 0.499i)4-s + (−0.964 − 0.265i)5-s + (0.395 − 0.101i)6-s + (0.559 + 1.72i)7-s + (−0.225 + 0.272i)8-s + (−0.141 − 0.301i)9-s + (−0.368 − 0.603i)10-s + (−0.753 − 0.707i)11-s + (0.252 + 0.139i)12-s + (0.417 + 0.886i)13-s + (−0.544 + 1.15i)14-s + (−0.427 + 0.388i)15-s + (−0.248 + 0.0313i)16-s + (−0.0558 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03266 + 1.31334i\)
\(L(\frac12)\) \(\approx\) \(1.03266 + 1.31334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 - 0.684i)T \)
3 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + (2.15 + 0.592i)T \)
good7 \( 1 + (-1.47 - 4.55i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.49 + 2.34i)T + (0.690 + 10.9i)T^{2} \)
13 \( 1 + (-1.50 - 3.19i)T + (-8.28 + 10.0i)T^{2} \)
17 \( 1 + (0.230 - 3.66i)T + (-16.8 - 2.13i)T^{2} \)
19 \( 1 + (-1.36 - 2.15i)T + (-8.08 + 17.1i)T^{2} \)
23 \( 1 + (-1.07 - 5.66i)T + (-21.3 + 8.46i)T^{2} \)
29 \( 1 + (8.22 - 3.25i)T + (21.1 - 19.8i)T^{2} \)
31 \( 1 + (-0.249 + 3.96i)T + (-30.7 - 3.88i)T^{2} \)
37 \( 1 + (-6.09 + 0.770i)T + (35.8 - 9.20i)T^{2} \)
41 \( 1 + (-1.87 + 9.82i)T + (-38.1 - 15.0i)T^{2} \)
43 \( 1 + (-0.905 + 0.657i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-5.66 - 6.84i)T + (-8.80 + 46.1i)T^{2} \)
53 \( 1 + (3.62 + 0.930i)T + (46.4 + 25.5i)T^{2} \)
59 \( 1 + (-6.38 - 3.51i)T + (31.6 + 49.8i)T^{2} \)
61 \( 1 + (-0.724 - 3.79i)T + (-56.7 + 22.4i)T^{2} \)
67 \( 1 + (7.01 + 2.77i)T + (48.8 + 45.8i)T^{2} \)
71 \( 1 + (-7.01 - 8.48i)T + (-13.3 + 69.7i)T^{2} \)
73 \( 1 + (-1.93 + 1.06i)T + (39.1 - 61.6i)T^{2} \)
79 \( 1 + (-6.10 + 9.61i)T + (-33.6 - 71.4i)T^{2} \)
83 \( 1 + (8.31 + 13.1i)T + (-35.3 + 75.1i)T^{2} \)
89 \( 1 + (-4.29 + 2.36i)T + (47.6 - 75.1i)T^{2} \)
97 \( 1 + (15.1 - 5.98i)T + (70.7 - 66.4i)T^{2} \)
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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

−11.10360126045996281710418092010, −9.233331464585410759548330456792, −8.703374020947371444392874129452, −7.972317253460399222238567525103, −7.32592166621661592996444734582, −5.88638389630715802515444904909, −5.53237304869199282508265181969, −4.18467100040787726234052828016, −3.16638840487613784623076863512, −1.89604793112101527206161093087, 0.70698368015840981079507615677, 2.66478471519813295631069515290, 3.68214701736771802526748487453, 4.46267024825253543852154774470, 5.11477003977485951394326911728, 6.79022091987252283879439631844, 7.59446295664580824736395309146, 8.161946218719484443958550075727, 9.566381273015217051824470886173, 10.38196625653107771072877517919

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