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Properties

Label 2-232-232.109-c1-0-22
Degree $2$
Conductor $232$
Sign $0.951 - 0.308i$
Analytic cond. $1.85252$
Root an. cond. $1.36107$
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  + (1.41 − 0.0799i)2-s + (2.11 + 1.01i)3-s + (1.98 − 0.225i)4-s + (−2.02 + 0.461i)5-s + (3.06 + 1.26i)6-s + (−2.62 − 1.26i)7-s + (2.78 − 0.477i)8-s + (1.56 + 1.95i)9-s + (−2.81 + 0.812i)10-s + (0.198 − 0.248i)11-s + (4.43 + 1.54i)12-s + (−0.296 − 0.236i)13-s + (−3.81 − 1.57i)14-s + (−4.73 − 1.08i)15-s + (3.89 − 0.896i)16-s − 2.23i·17-s + ⋯
L(s)  = 1  + (0.998 − 0.0565i)2-s + (1.22 + 0.587i)3-s + (0.993 − 0.112i)4-s + (−0.903 + 0.206i)5-s + (1.25 + 0.517i)6-s + (−0.993 − 0.478i)7-s + (0.985 − 0.168i)8-s + (0.520 + 0.652i)9-s + (−0.890 + 0.256i)10-s + (0.0597 − 0.0749i)11-s + (1.27 + 0.446i)12-s + (−0.0823 − 0.0656i)13-s + (−1.01 − 0.421i)14-s + (−1.22 − 0.279i)15-s + (0.974 − 0.224i)16-s − 0.542i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(232\)    =    \(2^{3} \cdot 29\)
Sign: $0.951 - 0.308i$
Analytic conductor: \(1.85252\)
Root analytic conductor: \(1.36107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{232} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 232,\ (\ :1/2),\ 0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47946 + 0.391949i\)
\(L(\frac12)\) \(\approx\) \(2.47946 + 0.391949i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0799i)T \)
29 \( 1 + (4.24 + 3.31i)T \)
good3 \( 1 + (-2.11 - 1.01i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (2.02 - 0.461i)T + (4.50 - 2.16i)T^{2} \)
7 \( 1 + (2.62 + 1.26i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-0.198 + 0.248i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.296 + 0.236i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
19 \( 1 + (4.05 - 1.95i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.12 - 4.94i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-9.44 + 2.15i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (-0.774 - 0.971i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 5.57iT - 41T^{2} \)
43 \( 1 + (-1.97 + 8.63i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-8.29 - 6.61i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.976 - 0.222i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 14.0iT - 59T^{2} \)
61 \( 1 + (4.65 + 2.24i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-6.36 + 5.07i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-3.19 + 4.00i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (12.3 + 2.81i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (7.67 - 6.12i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (-7.84 - 16.2i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (6.77 - 1.54i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-3.40 - 7.06i)T + (-60.4 + 75.8i)T^{2} \)
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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

−12.35210855701801379430007973785, −11.43438548668506546261576173524, −10.29005486049157701126871251956, −9.485894330203673788803057647637, −8.097051161455578529126799382480, −7.29590521513683947853148162588, −6.05820988170209526376000069695, −4.27918682412452811399628329303, −3.67300485749391678786969360324, −2.69043508459639760853344105883, 2.31576306534006108792061290213, 3.35013706891376569417571903467, 4.42551869733537056300134439063, 6.14431366922969301573803604949, 7.08442453827164984212806470322, 8.102406308280578395880816999956, 8.870470311567100082836126355159, 10.31278347079266983620746528228, 11.58645753853904604476605672663, 12.64722914747796574129237934989

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