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Properties

Label 1-1148-1148.23-r1-0-0
Degree $1$
Conductor $1148$
Sign $-0.640 - 0.767i$
Analytic cond. $123.369$
Root an. cond. $123.369$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.978 + 0.207i)5-s + (−0.5 − 0.866i)9-s + (−0.978 − 0.207i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.913 − 0.406i)23-s + (0.913 − 0.406i)25-s + 27-s + (−0.309 + 0.951i)29-s + (0.978 + 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.978 + 0.207i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.978 + 0.207i)5-s + (−0.5 − 0.866i)9-s + (−0.978 − 0.207i)11-s + (0.809 − 0.587i)13-s + (0.309 − 0.951i)15-s + (0.978 + 0.207i)17-s + (−0.104 + 0.994i)19-s + (−0.913 − 0.406i)23-s + (0.913 − 0.406i)25-s + 27-s + (−0.309 + 0.951i)29-s + (0.978 + 0.207i)31-s + (0.669 − 0.743i)33-s + (−0.978 + 0.207i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.640 - 0.767i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ -0.640 - 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04819572703 + 0.1029733093i\)
\(L(\frac12)\) \(\approx\) \(-0.04819572703 + 0.1029733093i\)
\(L(1)\) \(\approx\) \(0.6326661092 + 0.2088063650i\)
\(L(1)\) \(\approx\) \(0.6326661092 + 0.2088063650i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.978 + 0.207i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.5668262438185483268129712879, −19.66487860860340223941733863229, −18.93690948858591686539220919732, −18.49726841218694877748417577177, −17.56179251051642505592439162414, −16.80569447116254598142648278814, −15.84254535089870810535802005752, −15.52310509957867251572426775572, −14.13576972710257430010091401882, −13.51454804266705720047117006375, −12.62941609043735446600580239962, −11.99112086702436963253833044874, −11.28879811380718783322378286034, −10.60265344780108272194491774371, −9.40014998436626786183599318863, −8.234732808525436193832893796516, −7.80999400569678102427501513283, −6.99013686840642366360986849441, −6.05082801557691720176726974095, −5.15135851472794932422718289223, −4.264831621255113875903044176908, −3.129250024573887023019002276932, −2.06398669748995654078605571785, −0.88992898875933086929670852618, −0.03501281305295191146233245104, 1.07722629561575275478563899704, 2.85221393106086863676531810296, 3.58914849287067069923965258857, 4.298517803659179866622582378709, 5.41918383942124206077038693310, 5.963284920274882224378890192390, 7.189914137714501728524252731101, 8.17471385569766990815425427302, 8.65654430420215731300955510378, 10.16226763794570875603009598299, 10.40100224834712940874516926632, 11.26028605843309278805376351368, 12.129999685699311358565222912134, 12.69855833795577418240486425866, 14.03775168717379167486946731959, 14.74163106438711032678649033271, 15.707248854614959335073552946335, 15.9580196316512207904553094673, 16.74001092508449015651401913762, 17.76140885715299036401090301839, 18.544949144676871763751083609511, 19.17038991677511500299435178207, 20.48364431140508893878567121965, 20.624423333846665498221868086349, 21.58787293381232965319199106521

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