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Properties

Label 2-756-28.3-c1-0-4
Degree $2$
Conductor $756$
Sign $-0.391 + 0.920i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.577 + 1.29i)2-s + (−1.33 + 1.49i)4-s + (−3.11 + 1.80i)5-s + (−0.838 + 2.50i)7-s + (−2.69 − 0.861i)8-s + (−4.12 − 2.98i)10-s + (4.13 + 2.38i)11-s − 2.81i·13-s + (−3.72 + 0.365i)14-s + (−0.443 − 3.97i)16-s + (−1.10 − 0.640i)17-s + (−1.39 − 2.42i)19-s + (1.47 − 7.04i)20-s + (−0.694 + 6.70i)22-s + (−5.58 + 3.22i)23-s + ⋯
L(s)  = 1  + (0.408 + 0.912i)2-s + (−0.666 + 0.745i)4-s + (−1.39 + 0.805i)5-s + (−0.317 + 0.948i)7-s + (−0.952 − 0.304i)8-s + (−1.30 − 0.944i)10-s + (1.24 + 0.719i)11-s − 0.781i·13-s + (−0.995 + 0.0976i)14-s + (−0.110 − 0.993i)16-s + (−0.269 − 0.155i)17-s + (−0.320 − 0.555i)19-s + (0.329 − 1.57i)20-s + (−0.148 + 1.43i)22-s + (−1.16 + 0.672i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308446 - 0.466377i\)
\(L(\frac12)\) \(\approx\) \(0.308446 - 0.466377i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.577 - 1.29i)T \)
3 \( 1 \)
7 \( 1 + (0.838 - 2.50i)T \)
good5 \( 1 + (3.11 - 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.13 - 2.38i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.81iT - 13T^{2} \)
17 \( 1 + (1.10 + 0.640i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.39 + 2.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.58 - 3.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.78T + 29T^{2} \)
31 \( 1 + (-5.30 + 9.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.554 + 0.960i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.06iT - 41T^{2} \)
43 \( 1 - 9.45iT - 43T^{2} \)
47 \( 1 + (-2.18 - 3.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.06 - 5.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.957 + 1.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.59 - 4.96i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.12 - 4.68i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.12iT - 71T^{2} \)
73 \( 1 + (7.42 + 4.28i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.33 - 1.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.62T + 83T^{2} \)
89 \( 1 + (4.78 - 2.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.51iT - 97T^{2} \)
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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

−11.31259842157582100094230382281, −9.791470513649678873418697524507, −9.093320132879300885251408942617, −8.007602525976255661577999736226, −7.51342564535405817939255087350, −6.54420142448895504068439019576, −5.86554971015531778759457172504, −4.46953391897476858957675863265, −3.77682059983307310380601489926, −2.71907832642530805711597879603, 0.25796700476897096746972252943, 1.56168888877585882106209227251, 3.60207585613962888835823877106, 3.92908991623438865731544975479, 4.73535347435691114984411740723, 6.14060380287860271767495428586, 7.10350207362166373154636365827, 8.391825779749015179660297772365, 8.848041287473969916795046805316, 9.901845377376851442337523309853

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